Tips for fighting Battle Armor and Elementals

First, the most dangerous attack most BA/Elementals possess is swarming.  Remember which ones can swarm.  Elementals can always swarm...their specialties are swarming and steroids.  BA over 5t can't swarm.  Most, but not all, BA under 5t can swarm.  
Second, stay away!  When they're close, they can swarm you.  Don't get within 4 hexes and don't let them get within 4 hexes of you.  They're slow enough enough that you can often just stay away from them and deal with them after killing off everything else in the area. 
Third, if you get swarmed, remember what you're supposed to do if you catch on fire:  stop, drop, and roll.  Stop whatever you're doing, drop some swats on them, and do your special maneuver to shake them off.  You access this via hitting your active ability button on the toolbar at the bottom of the screen. Mechs use the "hand" icon for a hand swat, and next to it is an icon to roll over. Realistically, if you trust your pilot enough, intentionally baiting a swarming attack and then swatting them off is a good way to kill them. If you try this, you may want to do it toward the end of the mission in case the pilot starts bleeding. Also, try to get them to latch onto a vehicle instead of a mech if you can.  Erratic maneuvers are the vehicular equivalent of a swat/roll (and on the same menu) and, if successful, dumps them on the ground without evasion. The author can confirm this is a viable way of shaking someone off of a moving car as he's been on the receiving end. 
Fourth, EWAR is vital when it comes to BA/Elementals.  You want the best sensors and ECM you can get, since some of them have stealth you'll need to counter and, if you can see them at a longer range, you don't end up needing to get in swarming range to kill them. 
Fifth, you do have some long-range options, but early in the campaign you may not have access to them yet.  AOE weapons of any sort are great, since their evasion won't save them from an overpressure wave or a firestorm.  Also, any LRM or autocannon ammunition that says "FLAK" is your friend (and is also great for the destruction of pesky VTOLs).
Sixth, kill it with fire.  BAs and Elementals are like vehicles in that the take extra damage from fire.  So set them and the ground around them on fire, and keep doing it until the little thermometer pops out of their armor to let you know they're done. 
Seventh, get close to them.  I know, I know, but the best weapons against are short range weapons designed in part as anti-infantry (specifically, AP Gauss / Magshot; ER Micro Laser(C); Machine Guns; Machine Gun Arrays; Micro Pulse Laser (C); Micro Pulse Laser Array (C); Sm. ER Pulse (C); Small Pulse Lasers; and Small VSPL).  Those all get sizable bonuses to hit, and you wouldn't be reading this if you could hit those little bastards with anything else in your arsenal. Note that you'll probably need to stack some evasion ignore with the bonus to reliably hit them, though. If you choose this method, ensure that you've got enough of those weapons to properly damage your target. Even with boosted damage, those guns are still puny.
Eighth, the lead unit in a BA platoon counts as the "head" due to quirks in the coding, so you can't do a called shot on it unless you've a headshot-enabled mech. 
Ninth, B-Pods are Best Pods.  There's nothing quite like them for shredding a whole platoon in one glorious shot.  
Tenth, mines will not save you.  BA, Elementals, and mechs/vehicles weighing less than 20t cannot trigger mines directly, though they can be caught in the blast radius if someone else triggers them.  (Mine trigger chance table for reference.)
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