You can deposit contributions under the ESI Act, 1948, in the same way as you are doing for your Head Office employees located in Bangalore or Mumbai. The only difference will be that you will be required to make the contributions separately for the Mumbai Branch Office, quoting the necessary particulars of the said Branch Office and Sub-Code Number. You must submit the returns of contributions separately to the appropriate Regional/Sub/Divisional Office and the Branch Office of ESIC at Mumbai to which your Branch at Mumbai is attached.
However, at present, the online procedure for the payment of contributions, submission of returns, etc., has also been introduced. I think you can ask your Branch Office Manager/Incharge to contact the concerned Branch Office of ESIC at Mumbai to which your said Branch is attached and confirm all such formalities on the subject.
In addition to the payment of contributions for the Mumbai Branch Office employees, compliance regarding obtaining Insurance numbers, Identity Cards, Family Identity Cards, and submission of Accident Reports, if such an eventuality arises, etc., for Mumbai Branch employees must be done at the Mumbai office itself. Therefore, although you can pay contributions and submit returns from Bangalore, other formalities as mentioned above are required to be done locally at Mumbai so that the employees of your Branch Office may be able to get benefits under the said Act.