Ok I understand. Sounds like you need to use automatic archiving to move all your Gmail messages to the local folders in eM Client so they are safe when you remove the account.
Here is the eM Client blog on archiving and the various methods you can do.
https://www.emclient.com/blog/archive-your-emails-easily-with-em-client-315
Also see @Gary post below from the following thread https://forum.emclient.com/t/enhanced-archiving-feature/75035
“There is a difference between archiving and Automatic Archiving”
If your server permits it, an archive folder is created in the IMAP folders, and the archive button is an option in eM Client. This moves the message to the archive folder on the server which for Gmail accounts is the “All Mail” folder.
Automatic Archiving is something different. It runs unattended (hence automatic) in the background and moves messages based on age, from the server to local folders at the bottom of eM Client.