In order to migrate, I adjusted my old MX servers' preference, and added my new one; my goal being that the new one should be used, but if there's some problem, hopefully it's a problem early enough that it bounces and the old MX server is used instead.
This mostly worked. However the old MX server is still (after a couple of days) receiving some mail; consistently from the same single legitimate sender, and a load of spam.
When the old server receives this mail, it simply relays it to the new one, and it's accepted fine, so I think it's very unlikely it's actually being rejected in the first place.
I adjusted the priorities to 10 (new) and 65535 (old) in case more of a gap would make any difference; it didn't.
If it were just spam, I'd just ignore it as odd or even as a tactic - but why would the legitimate sender do this?