You'd think that any DSLR from, let's say, last ten years would be just fine, but it all just comes down to the internals of the cam in question. I'm just saying it's wise to check the experience on the other end before putting an old DSLR into production. Out of sync audio is more distracting than lower quality video.

Speaking of cheap cameras - don't count out older camcorders either. An older (but quality) camcorder will get you drastically better image quality and control capabilities than a webcam too, same process as hooking up an older DSLR (capture card, etc). Same challenges to be aware of with latency, and maybe moreso. I've seen a camcorder produced in last two years that had too much latency in HDMI signal to be worth the trouble - it was lower end, but just illustrates the point that component quality varies and has an impact on your use case.