The Indianapolis Children's Choir do this as a beat activity with 2nd and 3rd graders:

Children stand in a circle holding hands, they circle around in one direction on the first half of the verse (Riding in a buggy Miss Mary Jane, Miss Mary Jane, Miss Mary Jane). They walk the circle in the opposite direction for the second half of the verse (Riding in a buggy Miss Mary Jane, I'm a long way from home). Steps should be on the beat.

As the chorus begins the children drop hands, every second child will turn to face the child to his or her right, and every other child will turn to their left. The direction in which each child turns should be designated in advance. Each pair then does a 4-beat pat/clap pattern with their partner. Pat knees, clap hands together, pat hands against partner's hands, clap hands together again and repeat. (Who moan for me my darlin', who moan for me? Who moan for me my darlin', who moan for me?)

Children then hold hands and repeat from the start.