Here are videos that have been shared in previous umpire training sessions. These may be helpful as a refresher or as a reference for new umpires.
Putting the ball in play mechanic: used by the plate umpire when starting the game, starting every inning, after “Time” has been called, or if the ball becomes dead for any other reason
Called ball mechanic: used by the plate umpire when calling balls
Called strike mechanic: used by the plate umpire on a strike when the batter doesn't swing
Swinging strike mechanic: used by the plate umpire on a strike when the batter does swing, but doesn't make contact with the ball
Foul ball mechanic: used by the plate umpire when a ball is hit foul and not caught by any player
Check swing, batter offers mechanic: used by the plate umpire when a batter starts their swing and then attempts to stop it but is unsuccessful. It is also used when the batter attempts to bunt and does not make contact with the ball.
Check swing, ask for help - strike mechanic: used when a batter checks their swing and the catcher requests that the umpire asks their partner for assistance with the call.
Set position: as used by the base umpire prior to the pitcher delivering the ball.
Safe mechanic: used by both the plate umpire and the base umpire when making safe calls.
Set - safe mechanic: primarily used by the base umpire to come set prior to making a “safe” call so the play can be seen clearly
Out mechanic: used by both the plate and base umpire when making out calls
Set - Out mechanic: primarily used by the base umpire to come set prior to making an “out” call so the play can be seen clearly
Safe - ball is on the ground: used when it is not obvious to players, coaches, and fans that the fielder did not catch the ball resulting in the runner being called “safe”
Safe - off the bag mechanic: is used to let players, coaches, and fans know that the fielder pulled their foot off the bag resulting in the runner being called “safe”
Out - on the tag mechanic: used to indicate to players, coaches, and fans that the first baseman tagged the batter-runner resulting in the runner being called “out”