Yeah, it depends whether you have the old or the new Lighthouse. The new Lighthouse was published last year with the 2nd Edition of Seaside.
It also depends whether you have the old or the new Band of Misfits! The new BoM, which was published in 2019, doesn't emulate the Lighthouse. It just plays the Lighthouse from the supply. Image above.
If you have the new BoM...
Old Lighthouse - "while this is in play"If you play BoM on Lighthouse, BoM stays in play. But it doesn't protect you. The Lighthouse protects you if it's in play, but it's not (it stays in the supply); only the BoM is in play.
Playing Procession on BoM on Lighthouse doesn't make a difference. BoM is of course in the trash, but wouldn't protect you anyway.
New LighthouseIf you play BoM on Lighthouse, BoM stays in play. Lighthouse protects you because you played it (it doesn't have to be in play to protect you). It makes no difference where the BoM is. So in this case too, it doesn't matter if you trashed the BoM with Procession.
With both versions of Lighthouse, you always get the next-turn effects. Again it doesn't matter whether BoM was trashed or not.
Do you want to know how it would be with old BoM?