  A new era of ground-based observations, either in the infrared with the
next-generation of 25-40m extremely large telescopes or in the visible with the
8m Very Large Telescope, is going to be assisted by multi-conjugate adaptive
optics (MCAO) to restore the unprecedented resolutions potentially available
for these systems in absence of atmospheric turbulence. Astrometry is one of
the main science drivers, as MCAO can provide good quality and uniform
correction over wide field of views ($\sim$ 1 arcmin) and offer a large number
of reference sources with high image quality. The requirements have been set to
very high precisions on the differential astrometry (e.g. 50$\mu$as for
MICADO/MORFEO - formerly known as MAORY - at the Extremely Large Telescope) and
an accurate analysis of the astrometric error budget is needed. In this
context, we present an analysis of the impact of MCAO atmospheric tip-tilt
residuals on relative astrometry. We focus on the effects of the scientific
integration time on tip-tilt residuals, that we model through the temporal
transfer function of the exposure. We define intra- and inter-exposure tip-tilt
residuals that we use in the estimation of the centroiding error and the
differential tilt jitter error within the astrometric error budget. As a case
study, we apply our results in the context of the MORFEO astrometric error
budget.
