In a recent game I used Overlord to copy a Page just to get the cycling effect. The autoplay setting for Page was set to "Default is Exchange", so the game automatically exchanged the Overlord for a Treasure Hunter during Cleanup. That also happens if you have your Inheritance token on a Page and play an Estate.
The same thing probably also happens with Urchin, Settlers, Busting Village and may be relevant for inherited Estate-Crowns in your hand.
My opinion is that Overlord/Band of Misfits (BoM) should not 'take over' the autoplay settings of their copied card, because the copy is only temporary and my usual ideas about how to deal with played Pages are different from how to deal with played Overlords. OTOH, the change due to Inheritance is permanent for the game, so I think they should apply. If I want to keep my Estate-Pages (maybe for a Baron), I can set autoplay for the 1 inherited card to "off" for this game and decide on an individual basis.
What are your opinions on that?