Run up to tag up?

[voiceoflg]

A fan asked me this. Apparently some mistake us broadcasters for rules experts.

Runner on third, less than 2 out. Fly ball to right and the right fielder has a good arm. If the runner on third were to go a few steps toward second and get a running start to third, then tag third and head for home after the right fielder catches the ball, would that be legal? Instead of leaving from a dead stop, R3 would have momentum going and step on third just as RF catches the ball.

I said I had no idea but would find out. What say you?