The fastest way to transfer mail from your old provider to Paxmail is to use Thunderbird.
You can download it here https://www.thunderbird.net/en-US/

Login credentials for both the old mailbox and the new mailbox on our service.

A reliable internet connection (or enough free disk space if you plan to work fully offline first).

Open Thunderbird → Application Menu ☰ → Account Settings → New Account.

Enter the old mailbox credentials and let Thunderbird detect/confirm the IMAP (or POP) settings.

Repeat the same steps to add the new Paxmail address mailbox. Our server settings are:

Let Thunderbird finish its first sync so both account trees (Inbox, Sent, any custom folders) appear in the left-hand folder pane.

Tip: If you have many years of mail, give Thunderbird time; you can watch progress in Tools → Activity Manager.

Direct IMAP-to-IMAP copy
In the folder pane, select the folder (or highlight a block of messages) in the old account. 
For example, you might select the Inbox folder which contains the bulk of your messages.

Right-click → Copy To ↗ (or Move To ↗) and pick the matching destination folder under the new account.
You can also drag-and-drop, but the context-menu method is more reliable for large batches. 

Wait until Thunderbird finishes uploading; the green progress bar in the status line will disappear.

Repeat for every folder you want. For example, Sent, Archive (if you have it), and any other folders.

Open a few random messages in the new account to be sure body text, headers, and attachments display correctly.

If everything looks good, you may remove the old account from Thunderbird (Account Settings → Account Actions → Right click account and select (delete/remove)).

Finally, sign in to the Paxmail webmail to confirm the same folders and counts appear server-side.