Not yet reviewed by Pith; the record is open.

We develop and disseminate effective point-spread functions and geometric-distortion solutions for high-precision astrometry and photometry with the JWST NIRISS instrument. We correct field dependencies and detector effects, and assess the quality and the temporal stability of the calibrations. As a scientific application and validation, we study the proper motion (PM) kinematics of stars in the JWST calibration field near the Large Magellanic Cloud (LMC) center, comparing to a first-epoch Hubble Space Telescope (HST) archival catalog with a 16-yr baseline. For stars with G~20, the median PM uncertainty is ~13 $\mu$as yr$^{-1}$ (3.1 km s$^{-1}$), better than Gaia DR3 typically achieves for its very best-measured stars. We kinematically detect the known star cluster OGLE-CL LMC 407, measure its absolute PM for the first time, and show how this differs from other LMC populations. The inferred cluster dispersion sets an upper limit of 24 $\mu$as yr$^{-1}$ (5.6 km s$^{-1}$) on systematic uncertainties. Red-giant-branch stars have a velocity dispersion of 33.8 $\pm$ 0.6 km s$^{-1}$, while younger blue populations have a narrower velocity distribution, but with a significant kinematical substructure. We discuss how this relates to the larger velocity dispersions inferred from Gaia DR3. These results establish JWST as capable of state-of-the-art astrometry, building on the extensive legacy of HST. This is the first paper in a series by our JWST Telescope Scientist Team (TST), in which we will use Guaranteed Time Observations to study the PM kinematics of various stellar systems in the Local Group.