can I move a rollover IRA account with bought stocks, mutual funds from one company to another?

Lets assume that you have to sell the mutual funds and the stocks when going from financial company A to financial company B. If this was a taxable account, then you could face significant tax issues.

But this is a retirement account. The transfer has zero tax impact. The gains you would get if they are sold are either tax free or tax deferred. This is the case as long you also aren't going from non-Roth to Roth.

The key is not to cash them out, have the proceeds given to you, and then you send the proceeds to be new financial company. That type of exchange has deadlines. Contact the new financial company and they will help you do the transaction directly, with zero tax impact.

Now it is possible that the new financial company doesn't offer the same exact fund/class of fund that the old financial company does. In that case you have to determine if they have similar funds. But you should be able to re-purchase the shares of stock.