process H1 transfer while working in L1 status

[Aritra1911]

Hello,

I am presently in US working for my present employer A in valid L1. I also have an approved and valid till date H1B petition from employer B. The visa stamping is still due on this petition which was applied when I was in India. I only have the case number and one soft copy of I797 page with me. Is it possible to initiate an H1B transfer as lottery exempt against this petition, while I am in US in L1? If yes, how long will it typically take for the transfer process to take effect? How soon can I start working for my new employer sponsoring H1 transfer?

Thanks,

Aritra