My DIL in Ohio is wanting to move this plant to her place, but has no idea what it is or the best way to move it. Any thoughts?

It looks possibly like Bush Clover, maybe? Lespedeza

I just grew it for the first time this year so sorry, if that is what it is, I cannot really offer advice.

Yes. Probably one of the Lespedeza thunbergii cultivars, like Pink Fountains. You should be able to move it spring as it pops out of the ground or fall (now). Of course, if it's done in the fall, be sure to remove the flowers when transplanting.

I'd say move it just like you would any other shrub. I have three here and they're incredibly tough. Their roots may be the brittle type though, so maybe make sure the ground is nice and moist before digging.

Can you take cuttings, or divide it?

I've never tried it, but mine came as an itty-bitty piece of root with a shoot...so, maybe. Perhaps a slice off of an edge to see what happens?

Can't say for sure either, but when you dig it up, if it doesn't have a tap root, then yes. I assume it has at least a couple dozen stems.

According to one of the entries in the database, you can propagate it by division, so that's a good thing! Thanks so much for the prompt ID; she thinks that's most likely what it is.