Constant current in a circuit?
In electrical circuits velocity is pretty much the current (Coulombs/sec). Kinetic energy is proportional to velocity-squared (1/2*m*v^2), it means that if you have a constant current, you have on average a constant kinetic energy.

Therefore, since the entire wire is filled with electrons (virtually gapless); all the electrons must have the same velocity (same current), and so the kinetic energy  is equal every where.

Analogy, where water molecules = electrons. You can see that the molecules at the start of the pump don't have a larger velocity (current).


Another weaker analogy is that a train. Imagine the engine (battery) as the mechanism that applies the force (voltage/emf) to the rest of the carts (electrons). All the carts in the train will have the same velocity.