Martha Stewart Punches work perfectly with card (250-300gsm), even heavy embossed cardstock and most paper weights.
However they do not work very well with thin paper as the punch does not want to release the paper from the body of the punch. It is recommended that you use cardstock when you use them.
If they are going to be used with fairly thin paper, the paper should be inserted into the punch WITH some cardstock.
This should stop all the problems of the paper not being released.