There exists a standard stock transfer mechanism, ACATS, for basically "wiring" your stocks to a different brokerage. I think these transfers are also legally mandated to some extent.
My previous company partnered with ETrade, giving us a special account for stock awards. The main special thing about it were insane commission, order(s?) of magnitude higher than interactive brokers. I moved the stocks to IB using ACATS; it felt like ETrade tried to use stalling tactics like asking me to mail a paper form I think (I called bs), then iirc there was an error because the account was special so the transfer logic couldn't find it, etc. But after persisting I saved 100s of $$ and also the company had no control of the stock anymore (we had some earnings lock outs, not that it affected me).
The downside is that it might mess with automatic tax stuff, iirc dates-CBs got transferred but I had to redo something about these tx-es, maybe ISO stuff? Not sure if anything was missing for RSUs, I assume things like espp cost basis might get lost, we didn't have it.
Now I wonder if I should do the same for my current company stock... just in case.