As long as the current isn't heavy, they will. But, if there are creeks running off the river and there's shallows and cover, plus a low current, they'll go up in the creeks, as well. Even in moderate current, if there's an area of slack water (eddy) that's suitable for spawning, they're likely to use it. Mostly it all depends on where they were spawned, as they'll generally go right back to that same area when they're old enough to spawn.
That's based on what I've read, not what I "know" firsthand ... because I haven't fished rivers very much. I'm referring to "rivers" as like the KY River, which doesn't flow into a lake at any point .... and not like the Salt River (Taylorsville Lake), Tenn River (Watts Bar Lake), or Green River (Green River Lake), where I have caught lots of Crappie "in the main river" portion of the lake.