Travel-visa valid with employer A,working for employer B,want to change to employer C

I have a valid h1b stamped visa with A company until september 1st 2019, i moved to B company in 2018 and currently working with company B, and have valid i797 until nov 2020 with company B(but my visa Stamping is with Company A), i am travelling to india in august to do dropbox, meanwhile i got another offer and want to transfer h1 to Company C, and want to move to company C. 1. If I go to india and while doing dropbox with company B, company C asks for documents for h1b transfer, is it OK to do that? 2. If I go to india and while dropbox with company B is in progress, company C has already initiated my h1b transfer., is this scenario OK?
3. Is it necessary to do a dropbox now with company B (current company)? since i already have valid visa until sep 1st 2019 with company A?

It is absolutely fine as you are on H1B and others can apply for transfer…You should go for dropbox using the company that you plan to work for and that you have approval of H1B with.