Swing the opposite arms and legs in sync while running.
Pump the arms forward and backward in line with the direction of movement. The arms should not swing across the body and the elbows should point backwards, not outwards.
Swing the arms from the shoulders, not the elbows. Keep the elbows bent throughout the swing.
Swing the arms through a full range of motion. In general, the hands should move from about the hip to the chest.
Keep the shoulders and hands relaxed. The hands shouldn't be clenched into fists, but they should be stable, not flopping around. The shoulders should be down and back, not high and tight. This along with bent elbows will allow the hands to pass the body at about hip height.
Do not worry about having kids perfect these movements. Focus on increasing their body awareness and control, as well as developing their appreciation of efficient, purposeful movement. It should be fun to learn how to move skillfully.
Look at the activities featured in Running Start's Coaching Videos for exercises that will develop these arm movement essentials.