When a player decides to drop an investigator and start a new deck, what happens with their supplies? Does it stay with the new investigator or do you lose them?

Supplies are associated with an investigator, so if that investigator doesn't continue for whatever reason (insane, killed, just didn't want to play with) the supplies are lost.

Okay,but do we have any text backing up our interpretation of what happens?

Yeah, that's been confirmed by Matt, but we had no ideal place of adding that to arkhamdb or another repository of rulings. For what it's worth, he said it would probably be addressed in the next FAQ.