At the very least keep the chain and cassette the same speed, yes. I'd just keep all the same-speed stuff together. There's a lot of times it all works in a gross sense but the little details just aren't good. Like, pretty much any chain and double or triple crankset can go together but the front derailleur will work best shifting the crankset and chain width it was designed for, both in chain width, ring spacing, and tooth jump. An 11 speed rear derailleur doesn't care about chain width but it might run out the available cable pull from a 7 speed friction shifter. Got to think about cage length on the RD too, vs both chain capacity and cog capacity.
Don't forget to check you have enough chain length on your new bike.
__________________
Genesis 49:16-17