Where does the energy needed for the movement of molecules in diffusion and osmosis come from?
Heat, pressure, friction, sound, kinetic energy of other molecules, etc.
Kinetic energy comes from all sorts of places, as all energy is really the same. It can come from sound waves, heat, friction and pressure, or even the kinetic energy of other molecules moving around and bumping into each other.
You can think of osmosis and diffusion more as laws governing the direction of movement, while the actual movement itself is caused by energy from the environment.
If there is no energy in a system, then osmosis and diffusion won't happen. If there is energy, the molecules will tend to move in directions according to osmosis and diffusion. And energy comes from almost anything.