In reply to:
..Easyspace failed to remind me to renew and slapped a £3 charge for each service that was not renewed so I just want to transfer away without paying them anything.
Well, if you have already let it expire, you probably will not be *able* to "transfer" it away, I don't know about EasySpace in particular, but most registrars will *not* xfer a domain that is in expired status. Some will let you "redeem" it and renew it during the "grace period" after expiration (sometimes for an additional fee) while others just pass it on to their "backorder pool". At the end of the grace period, it should be returned to the available pool ... though with backorder systems and stuff, it may never be.
In reply to:
The domain no longer shows up on my control panel so presumably it will expire in the next few months if I just leave it.
It's either expired or it hasn't ... and it may or may not be in "grace period".
If you did not pay EasySpace for a renewal unitl it "expired" with them, you are probably going to have to "redeem it" with them from within the grace period, or just wait to see if it becomes available for general registration again.
If it does become available for registration again, then there is no "transferring" involved at all ... just register it with DH or any other registrar.
BTW, you should *never* rely on *anyone* to "remind you" to renew your domains. (but I guess you have figured that out already). 
--rlparker