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Let me answer, I'm sure Leigh will as well.
The Total kinetic
energy is what I would call the amount of Heat in each container. Heat is
energy but not all energy is heat--stuck with a dual nomenclature due to
the caloric sidetrack in history.
Now when A and B are placed in contact I watch over time and find that the
motion in A decreases while the motion in B increases. Eventually, the
total KE in A equals the total KE in B. Energy has been transferred from A
to B and it took time to do so. This has all the earmarks of a 'flow' from
A to B.