Suppose I have a Blackbird-equipped Cyclops (Slim,) then I play another Cyclops (Scott Summers,) since the older Cyclops has to leave play, does the equipment go with him or does the freshly recruited Cyclops take over the equipment?

nope. equipment goes bye bye.

Sadness. :(

Thanks for the clear up. :)

If it was any other way, I suspect most decks would pack lots of equipment. Mainly personal forcefield. thats just becomes ridiculous.