Magnetic fields can be crucial in guiding the mass assembly of star-forming gas in molecular clouds. Joint studies of magnetic fields and gas kinematics on filament scales, however, are rare. Here, we present one of the first relative orientation measurements between magnetic fields and velocity gradients in a globally collapsing, cluster-forming complex. Specifically, we analyzed the dust polarization measurements of the Perseus NGC 1333 taken by the JCMT BISTRO survey at a 0.02 pc resolution, and the NH3 spectral line information from the GBT+VLA combined data, convolved to the same resolution. While we did not find a global preferential alignment between the inferred magnetic field and gas velocity gradients, we found local alignments between them that are preferentially perpendicular and parallel at various locations. We also found analogous alignments between magnetic fields and column density gradients locally. These results indicate that magnetic fields in star-forming clusters can be quite complex on the < 0.1 pc scales, likely due to distortion by global and local contractions of the star-forming gas or proto-stellar feedback. The local magnetic fields, however, can remain significant in guiding accretion flows on < 0.1 pc scales under certain physical conditions, as indicated by the localized preferential alignments between magnetic fields and velocity gradients in some locations.