Snap caps are intended to cushion the firing pin so that it does no damage to itself or the surrounding part of the pistol. They will not cycle the pistol; in your words, they will not pop out on their own.
If your pistol has a hammer, you can fire again on the same snap cap either double action (pulling the trigger without cocking the hammer) or single action (cocking the hammer with your thumb before pulling the trigger).
If it is a striker type pistol without a hammer, you can usually cock the firing mechanism with a small movement of the slide that comes short of ejecting the snap cap. In either type, you can also cycle the slide fully by hand to eject the current snap cap and load the next one from the magazine.
I hope that helps.