Do freefalling clocks within a small region of spacetime undergo gravitational time dilation? I have several times seen explanations of gravitational free fall (eg, of a small object toward earth with no air resistance) that begin with the following claim about the particles of the free-falling object: its particles that are closer to earth are experiencing greater gravitational time dilation than are the particles that are further from earth.  Example of this claim used in explaining  gravity: https://youtu.be/UKxQTvqcpSg
I understand that if a clock is at rest at a fixed distance from earth's center, then the clock exphibits greater time dilation the closer it is to earth's center.  But the latter scenario is not free fall.  And to me, the aforementioned claim seems contrary to the equivalence principle. 
I'll put my question this way: Suppose you have a free-falling lab within a small region of spacetime, at an altitude of several km, with no air resistance, and there are two free-falling clocks in the lab, one of them a few nm closer to earth than the other.  Will the freefalling clocks undergo gravitational time dilation relative to one another such that an observer in the lab will observe the clocks to be ticking at different rates?
I have no formal education in physics, and I would prefer an answer that is more conceptual and less mathematical, because I may not be able to follow the math very far.
 A: The equivalence principle dictates that objects in freefall in a gravitational field are not considered to be accelerating.  To the extent that the gravitational field is uniform, two clocks, one above the other, will measure time at the same rate.  This rate will be slower than a reference clock far above and outside the gravitational field.  
In practice, the field may not be exactly uniform. For example, free-falling towards the Earth, the field might be very slightly higher on the lower of the two free-falling clocks. This is a 'micro-gravity' or a tidal effect.  In this case, the lower clock would run very very slightly slower. This is not due to the gravitational field and resulting difference in potential since this is cancelled by the freefall acceleration common to both clocks. It is due to the gravitational field gradient (i.e. the fields and thus the effective potentials are not exactly the same on the two clocks). For practical separations, this difference would not be measurable.   
[Edit] My comment "the lower clock would run very very slightly slower" is incorrect.  It could run slower or faster.  Using a clock placed at the center of mass of the free-falling (or orbiting) laboratory as a reference, clocks placed above or below this position will run more slowly.  The gravitational potential near a planet is concave downwards.  For exactly the same reason, there are two ocean tides each day, not one.
A: If you intend to place the clocks very close to each other, it does not make sense to speak about the gravitational time dilation. However, if you place the freely falling clocks a meter or a kilometer away from each other, the time dilation becomes tangible. Since the gravitational time dilation depends on the gravitational potential of the point at which the clock is located, the time rates of the clocks would be different from the viewpoint of the observer located on the earth whether or not the lab is fixed at a large altitude from the earth or is freely falling at that point.
However, when the lab is fixed, there are solely different gravitational time dilations for the clocks due to different potentials, whereas for a freely falling lab, besides the said time dilation, there is an additional time dilation due to the instantaneous speed of the lab (clocks) relative to the terrestrial observer, which is an SR effect.
If you want to compare the clocks from the standpoint of the lab observer, both the SR and gravitational time dilations seem to happen for the clocks, and the SR effect mainly arises from the tidal forces that tend to accelerate the clocks WRT each other as well as the lab observer.
A: Time dilation does not depend upon acceleration; this is has been experimentally verified in particle accelerators. So whether the clocks are undergoing acceleration or not does not affect their rates relative to distant clocks.
The answer to your question therefore is that clocks closer to the center of the Earth undergo time dilation relative to clocks further away from the center of the Earth, and this is true whether the clocks are in free fall or not.
