Q:

What processes exist to issue formal warnings or remove moderators? 

Feedback and questions about these processes are on this other post. 

This post lists two process by which formal complaints against moderators are handled by the Community Management Team. 
The Moderator Action Review Process is a process for allowing a team of moderators to remove one of their own.  This process is for those rare situations where communication with one member has completely broken down and the team as a whole feels they cannot continue to work together. 
The Moderator Conduct Review Process is a process by which complaints against a moderator's conduct can be raised.  This process is for when there's reason to believe the Code of Conduct or moderator agreement have been broken, or if a moderator's privileges need to be immediately revoked in an emergency (e.g. if their account appears to be compromised). 
In addition, there also exists a third process for removing moderators who have been inactive in moderating their sites.  This process is for when a moderator hasn't performed a moderator action in several months, in order to ensure that only those who actually use the moderator tools have access. 

A:

Moderator Conduct Review Process:
From time to time, a moderator on one of our sites may violate their moderator agreement, or the company standards for behavior as documented in our Code of Conduct.  Such behavior may eventually necessitate their removal from office, however both community expectations and our Code prescribe that, in general and whenever possible, they first be warned and given an opportunity to correct their behavior.  Clear, actionable feedback and guidance are essential tools in helping moderators understand and enforce our Code of Conduct, and this feedback should always be the first option if the moderator could have simply failed to perceive their actions as problematic. 
This process also encompasses Emergency Removals: these can involve scenarios such as evidence of a moderator’s account being compromised or the moderator acting detrimentally and extremely out of character, a moderator violating their moderator agreement (by sharing PII, for instance), client-side scripts causing problems, general security incidents, and other unanticipated incidents that put the immediate safety and trust of our users and communities at risk.  

Initiation
The process is started when the Community Management Team is made aware of an issue.  This can be a complaint against a moderator, or the Team being made aware of an issue causing immediate harm (security issue like a moderator's account appearing to be compromised, or moderator agreement violation as described above).  In the latter case, the Emergency Removal Phase is initiated; in the event of any other complaint against a moderator's actions, that step can be skipped, and the Discovery Phase will be the starting point.    
Any time moderator access is removed, the rest of the team(s) to which the moderator was a part of will be made aware of the situation too.  Communication surrounding the removal with the community is handled in a similar fashion suspension are. 
Emergency Removal Phase

The CM (CM1) who "stumbles upon" the issue will take care of it. 
Moderator access should be removed immediately, to limit the potential for any further damage.  Available info is documented. 
CM1 will reach out to the moderator to let them know of the situation.  Other moderators on the team will be notified too. 
Skip to Discovery #4. 

Discovery Phase

Concern is raised by user, fellow moderator, or staff.   
CMs can recuse themselves if they feel they can't be impartial.  CM1, CM2, and CM3 will be picked at random from remaining CM pool.   
CM1 will review data and research concerns - look through account annotations, query the involved parties, etc.  If there is a note to check with the Community Strategy Team (CST), they'll do so: the CST may have information relevant for next steps.   
All the info should be compiled by CM1 into a doc, along with their conclusions:  

If CM1 determines no CoC or moderator agreement violation occurred:  

If an Emergency Removal did not take place, the process is concluded here.   
If an Emergency Removal took place and it's since been explained, and no further action is needed, moderator access can be restored, and any appropriate security measures should be communicated.  The process is concluded here.  

If CM1 determines a CoC or mod agreement violation occurred, they will contact the moderator about whom the complaint was raised and inform them of the ongoing process, prompting them for their perspective/context/reasoning, and add this to the documentation to be reviewed in the Confirmation Phase. 

Confirmation Phase

CM2 will need to confirm CM1's findings.   

If CM2 disagrees with CM1's findings, this will be added to the documentation along with their findings.     

If CM2 is able to convince CM1 that no violation has occurred, they will document their findings, and provide feedback to the moderator.  If this was an Emergency Removal, access is restored and security measures communicated too (any errant annotations are removed from account).  The process is concluded here.   
If CM1 and CM2 can't reach a consensus, CM3 will break the tie.  Confirmation #1.1.1 or Confirmation #1.2 must ensue. 

If CM2 agrees with CM1's findings, CM2 will proceed to the Execution Process. 

Execution Phase

If an Emergency Removal took place, moderator access can't be restored.  At CM2's discretion, the network account may be suspended for 30 days to prevent further harm while issue is being handled.  CM2 will communicate their findings to the moderator, and annotate their account.  The process is concluded here.  
CM2 will look through the moderator's account, for annotations.   

If there's is no annotation that notes the moderator had previously been warned about the same issue, CM2 will do so (along with suggestions for adjusting behavior), and annotate the account.  The process is concluded here.  
If there is an annotation that notes the moderator had previously been warned about the same issue, CM2 will notify the moderator that their access will be removed and explain why, and they will annotate the account.  The rest of the moderation team(s) is also to be informed of this.  The process is concluded here. 

Here's a flowchart of the process (click to expand; high-contrast version):

A:

Moderator Action Review Process
After much discussion, debate, reflection, consultation, frustration, revelation, constipation  and inspiration, we've settled on the following process for allowing a team of moderators to remove one of their own.  It is, by necessity, somewhat formal: this process is for those rare situations where communication with one member has completely broken down and the team as a whole feels they cannot continue to work together. 
I strongly urge anyone involved in such a situation to do what they can to resolve outstanding issues before resorting to this. 
Initiation
The process will be initiated by a formal request from one or more moderators on a site, sent privately via an email to community@stackexchange.com. 
The process may also be initiated by the Community Team at Stack Exchange, Inc. in response to numerous, substantiated complaints from users on the site. 
In either case, the complaints will be treated by Stack Exchange as confidential, and their authors will not be named by us at any point in the process. 
Proceedings
Once begun, the following steps must be followed, in order, to completion within a reasonable time frame.  If this is not possible, all participants will be notified by us that the process has been discontinued and informed of the resolution (if any). 

All moderators on the site will be contacted by us via email, informed of the situation and asked to meet at their earliest opportunity to discuss the removal of the named moderator.  Meeting must be held in a private venue to maintain the confidentiality of those involved — we will provide a chat room on http://chat.meta.stackexchange.com that is inaccessible to anyone not invited.  Both the venue and timeframe for the meeting must be accessible to all moderators — we will attempt to coordinate the schedules of individual moderators. 
At the designated time, a quorum must be present — this shall consist of &frac23rds (rounded up) of the moderators on the site (all those listed as active on the /users?tab=moderators page, whether or not currently active), excluding the moderator to be removed (example: for a team with three moderators, both of those not being considered for removal must be present). 
A designated person will be selected to record the minutes of the meeting.  These should be brief, and suitable for public consumption should the need arise (providing only a broad overview of the process and its outcome, not including any details of what was discussed). 
At this point, each present member of the moderator team shall be given an opportunity to share their concerns with the group.  In the event that this process was initiated by complaints from outside the moderator team, a summary of them will have been provided to the moderators prior to the meeting. 
Following this, the moderator to be considered for removal shall have a chance to respond.  There will be no back-and-forth discussion allowed — this should have been conducted prior to this meeting. 
Finally, the moderator to be considered for removal shall be asked to leave the room (upon which access to the chat room shall be revoked) and otherwise remain silent through the remainder of the proceedings, and those remaining shall vote on whether or not to revoke the moderator's privileges. 
If at least &frac23rds of those present vote for removal, this shall be considered a consensus, and recorded in the minutes as the opinion of the moderator team. 
The meeting shall now be concluded, and the minutes emailed to all members of the moderator team and community@stackexchange.com. 
If the consensus was for removal, we will then revoke the privileges of the moderator to be removed, and also remove the moderator's name from the election and /users?tab=moderators pages. 

How — or if — the outcome of such a meeting is shared with the broader community will be left to the discretion of the moderator team.  However, the details of the meeting must remain confidential — only the minutes can be published if such a need arises.  This is done to allow potentially-confidential information to be disclosed without forcing anyone involved to violate the moderator agreement. 
In general, the same courtesies should be extended toward removed moderators as to suspended users: no airing of dirty laundry in public, no bringing up issues faced as a moderator in unrelated discussions, questions raised by other members of the public answered with as little detail as possible.  Speculation should be discouraged out of respect for those involved. 

This has been copied from an older post here.