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The Planck length is only a feature of the quantum theory. It is the distance beyond which the quantum theory cannot resolve separate locations. It doesn't say anything motion in nature. If space is a continuum, which I believe it is, then motions on scales below the Planck length are definitely allowed.
Overall, position states in Hilbert space describe the state of being between two locations, not the state of being at any one location. Due to the Heisenberg uncertainty, we can never squeeze the boundaries of a position state closer together than the Planck length but we have good reason to think that the boundaries of one position state might be shifted by less than one Planck length with respect to another. I don't know of they could exist in the same Hilbert space but they would describe the same position space without discrepancy. To do so, just define a coordinate system with its origin shifted by less than one Planck length from another. Since no coordinate system is favored, the continuum of possible coordinate systems suggests a continuum of positions.
Pic related; If Psi_AB and Psi_CD are states such that a particle is between A and B and C and D respectively, then the distances AB and CD must be at leas the Planck length. The distance XC, however, can be arbitrarily small.
