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The Extremely Large Telescope and the Thirty Meter Telescope will use state of the art multiconjugate adaptive optics (MCAO) systems to obtain the full D4 advantage that their apertures can provide. However, to reach the full astrometric potential of these facilities for on-sky science requires understanding any residual astrometric distortions from these systems and find ways to measure and eliminate them. In this work, we use Gemini multiconjugate adaptive optic system (GeMS) observations of the core of NGC 6723 to better understand the on-sky astrometric performance of MCAO. We develop new methods to measure the astrometric distortion fields of the observing system, which probe the distortion at the highest possible spatial resolution. We also describe methods for examining the time-variable and static components of the astrometric distortion. When applied to the GeMS Gemini South Adaptive Optics Imager (GSAOI) data, we are able to see the effect of the field rotator at the subpixel level, and we are able to empirically derive the distortion due to the optical design of GeMS-GSAOI. We argue that the resulting distortion maps are a valuable tool to measure and monitor the on-sky astrometric performance of future instrumentation. Our overall astrometry pipeline produces high-quality proper motions with an uncertainty floor of 45 uas per year. We measure the proper motion dispersion profile of NGC 6723 from a radius of 10 arcsec out to 1 arcmin based on 12000 stars. We also produce a high-quality optical-near infrared color-magnitude diagram, which clearly shows the extreme horizontal branch and main-sequence knee of this cluster.