  Time-evolving random graph models have appeared and have been studied in
various fields of research over the past decades. However, the rigorous
mathematical treatment of large graphs and their limits at the process-level is
still in its infancy. In this article, we adapt the approach of Athreya, den
Hollander and R\"ollin (2021+) to the setting of multigraphs and multigraphons,
introduced by Kolossv\'ary and R\'ath (2011). We then generalise the work of
R\'ath (2012) and R\'ath and Szak\'acs (2012), who analysed edge-flipping
dynamics on the configuration model -- in contrast to their work, we establish
weak convergence at the process-level, and by allowing removal and addition of
edges, these limits are non-deterministic.
