The protoplanetary disk around MWC 758 is a source with complex substructure including an inner cavity, spiral arms, two large dust clumps, and a dust ring with eccentricity ~0.1. This project consists of two parts: (1) studying the gas kinematics of 13CO and C18O to constrain the gas eccentricity, and (2) analyzing 1.3 continuum data to measure the proper motion of the two dust clumps. In the first part, we obtained non-Keplerian velocity deviations by subtracting an ideal Keplerian model from the observed velocity pattern. We examined several possibilities that may cause the velocity deviations and found that eccentric orbital motion with eccentricity 0.1+-0.04 best explains our results. The gas eccentricity is consistent with that of dust, indicating strong dust-gas coupling. For the second part, we found that overall the dust clumps follow the Keplerian flow, which is consistent with the theoretical expectation if the clumps are dust-trapping vortices. Nevertheless, hints of deviations from Keplerian speed are seen, which may be due to other mechanisms such as spiral motion or radial flows.