Legal.
Unless for some other reason unrelated to the checked swing appeal, the ball remains live while the appeal is being made. Runners may advance- and be put out!- during this live ball.
Most every time this appeal is made it will be during a live ball situation, just a few seconds after the pitch has arrived. The defense needs to be ready for any advancing runners.
I can envision at few times that the appeal would be made during a dead ball, thus keeping the runners on their bases. One example: Pitcher throws, batter checks swing and is then hit by the pitch. Whenever a pitch touches the batter, you have a dead ball and no other plays can be made.
The defense could then appeal the checked swing without worrying about the runners. If appealed and ruled a swing, you have a strike on the batter, and an out if strike three. If the base umpire rules "no swing", then you would have a possible award of first base for the hit-by-pitch.
Another example would be a checked swing on a pitch that goes out of play, which also becomes a dead ball. We could probably think of a couple more, but the times a checked swing appeal would be made on a dead ball are few and far between. Almost always this appeal is made while the ball is live and runners can advance.
One way to stop the runners: Quickly fire the ball back to the pitcher before making the appeal. The Look Back rule will then be in effect and that should freeze the runners or force them to commit to a base.