In "chemical" reactions we usually have some sort of electron orbital change. We also have "thermal" reactions, in which the kinetic energy (energy of motion)of the molecule can be gained or lost (velocity speeded up or slowed down.)
As for the "motion" one, remember that Einstein said that as objects approach the speed of light they get infinitely massive. Well, they gain mass at slower speeds too -- the mass gain is due to the energy associated with motion. (Relativity involved here.)
I suppose it is harder to see the mass gain/loss of electron orbital changes -- but they usually involve imparting or gaining energy from the motion of the particle, or the emission or recption of a photon (light.)
You can see that photons carry off energy -- and that's also your mass loss.
So an exothermic reaction (chemical or thermal) carries away mass/energy in either emitted photons or the transfer of motion energy to a neighbor molecule through collision.