H1B Transfer while H4 EAD application pending

My wife has stamped H4/I94 until Dec 2022 and we have applied for her EAD recently. Now, I have got a new job and need to transfer my H1B. The new employer is saying that I will need to file an H4 for my wife else she will be out of status. Have few questions :

1 - Is it correct that H4 has to be transferred along with H1, though the H4 is stamped for 3 years already using old H1 petition.

  1. If we transfer the H4/H4EAD along with H1 tranfer, what happens to the pending EAD application. H4 extension is taking longer and we need the EAD sooner.

Please help.