Question: Can the bank ask me to give KYC documents only at the same branch if I transferred my account?
Answer: Not necessarily. RBI allows updation at the home branch or another branch of the same bank, subject to policy.

Supporting evidence:
title: rbi.org.in/notification.aspx
quote: KYC verification once done by one branch/office of the RE shall be valid for transfer of the account to any other branch/office of the same RE, provided full KYC verification has already been done for the concerned account and the same is not due for periodic updation
context: reaches rupees forty thousand or the total credit in a year reaches rupees eighty thousand that appropriate documents for conducting the KYC must be submitted otherwise the operations in the account shall be stopped when the total balance in all the accounts taken together exceeds the limits prescribed in direction (d) and (e) above.
81
The account shall be monitored and when there is suspicion of ML/TF activities or other high-risk scenarios, the identity of the customer shall be established as per
paragraph
16 or
paragraph
18.
25.
82
Deleted.
26.
83
KYC verification once done by one branch/office of the RE shall be valid for transfer of the account to any other branch/office of the same RE, provided full KYC verification has already been done for the concerned account and the same is not due for periodic updation.
Part II - CDD Measures for Sole Proprietary firms
27.
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For opening an account in the name of a sole proprietary firm, CDD of the individual (proprietor) shall be carried out.
28.
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In addition to the above, any two of the following documents
or the equivalent e-documents there of
as a proof of business/ activity in the name of the proprietary firm shall also be obtained:
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Registration certificate
including Udyam Registration Certificate (URC) issued by the Government
Certificate/licence issued by the municipal authorities under Shop and Establishment Act
Sales and income tax returns
87
CST/VAT/ GST certificate
Certificate/registration document issued by Sales Tax/Service Tax/Professional Tax authorities
IEC (Importer Exporter Code) issued to the proprietary concern by the office of DGFT or Licence/certificate of practice issued in the name of the proprietary concern by any professional body incorporated under a statute
Complete Income Tax Return (not just the acknowledgement) in the name of the sole proprietor where the firm's income is reflected, duly authenticated/acknowledged by the Income Tax authorities
Utility bills such as electricity, water, landline telephone bills, etc.

29. In cases where the REs are satisfied that it is not possible to furnish two such documents, REs may, at their discretion, accept only one of those documents as proof of business/activity.
Provided REs undertake contact point verification and collect such other information and clarification as would be required to establish the existence of such firm, and shall confirm and satisfy itself that the business activity has been verified from the address of the prop