Apple Mail not syncing moved emails to IMAP server

I switched from an email provider to another, so I moved e-mails from the Apple Mail account of provider 1 to the mail account connected to the provider 2.

All my e-mails appear now in the account linked to provider 1, but on the server, only new emails are stored (i have access to the Maildir directory server side). All my previous (moved) emails are not stored on the server.

From the account information in Apple Mail, I see the tab Quota, it reports all IMAP directory empty (except new inbox which received new emails since the move). And reports a size of 0. But when using the button "Show the messages", it shows the directory containing my old mails.

So, I understand that Apple Mail stores only locally, and do not synchronise back to the server. Is it possible to force sync to server ?

Thanks to help me understand how Apple mail really works

Christophe

[Re-Titled by Moderator]

Original Title: Apple Mail do not store on imap server when moving e-mails from another account. But mail is still shown in the new imap account