No, the firearm doesn't make the bullet tumble. But, it can be designed so that the bullet is more or less likely to tumble on impact.
The rifling can impart just enough spin to the bullet to keep it stabilized for a short flight, but tumble on impact. But, this results in decreased accuracy at longer ranges.
Conversely, A higher spin rate stabilizes the bullet for longer distances, but inhibits tumble on impact.
The round tumbles when it hits an object, the reason being a pocket of air left in the nose of the bullet. The nose collapses, the bullet destabilizes, and rips itself apart as it travels through an object.
You are describing fragmentation of a hollow-point bullet, not tumble. The later may happen as a consequence, but tumbling will occur even with a full-metal-jacket under the conditions I described above.
some fmj mil ammo have hollow noses - the core moves forward on impact to cause tumble, not a hollow point as that is against the geneva conv.