Question

what is transferred when moving things run into each other?

Answer 1

kinetic energy.

Answer 2

Although kinetic energy is not traded. Like, one object might lose 50J and the other will gain 30J. It depends on the collision!

If I remember correctly, though, momentum IS traded.
Thus, you can rely on the "conservation of momentum!"

What about conservation of energy? Well, energy is conserved. But mechanical energy might not be conserved. So we can't rely on it unless we assume the collision is "ellastic."