The Gemini multiconjugate adaptive optics system (GeMS) is a facility instrument for the Gemini South telescope. It delivers uniform, near-diffraction-limited image quality at near-infrared wavelengths over a 2 arc min field of view. Together with the Gemini South Adaptive Optics Imager (GSAOI), a near-infrared wide-field camera, GeMS/GSAOI's combination of high spatial resolution and a large field of view will make it a premier facility for precision astrometry. Potential astrometric science cases cover a broad range of topics including exoplanets, star formation, stellar evolution, star clusters, nearby galaxies, black holes and neutron stars, and the Galactic Centre. In this paper, we assess the astrometric performance and limitations of GeMS/GSAOI. In particular, we analyse deep, mono-epoch images, multi-epoch data and distortion calibration. We find that for single-epoch, undithered data, an astrometric error below 0.2 mas can be achieved for exposure times exceeding 1 min, provided enough stars are available to remove high-order distortions. We show however that such performance is not reproducible for multi-epoch observations, and an additional systematic error of similar to 0.4 mas is evidenced. This systematic multi-epoch error is the dominant error term in the GeMS/GSAOI astrometric error budget, and it is thought to be due to time-variable distortion induced by gravity flexure.