mailkodo doesn't have a feature that automatically pulls mail from your old mailbox. But the IMAP protocol itself supports opening two mailboxes on two different servers at once and copying messages by hand — and because mailkodo speaks standard IMAP, this path is always open.
Most desktop mail clients, Thunderbird among them, show multiple accounts side by side. Add both your old mailbox and your mailkodo mailbox to the same client as IMAP accounts, and you can drag and copy mail between folders.
Both mailboxes stay open during the move — you decide when to close the old one, mailkodo doesn't do that for you. It's slower than a one-click automation, but it gives you full control: you decide which folder, when, and how much to move.