Probably you meant that there's possibility of slight difference between gravitational and inertial masses. IIRC it's based on the definition that full energy E = mc^2 + K + W, there K - kinetic energy, and W - potential energy in the gravitational field, so higher module of W (closer to gravitating mass) means lower E (and smaller mass), since W is negative.
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He also repeatedly claims that the "twin paradox" and the atomic clock experiment relating to the effect of time dilation is logically faulty because movement is relative. However, in standard science, it is not movement that is responsible for the time dilation effect but acceleration.
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Actually t'= t*sqrt(1-v^2/c^2). There's nothing about acceleration here. Accelerating frames are matter of special relativity. Also twin paradox can be solved in special relativity only (edit: sorry not special, but general relativity in both cases).