  The competition between scrambling and projective measurements can lead to
measurement-induced entanglement phase transitions (MIPT). In this work, we
show that the universality class of the MIPT is drastically altered when the
system is coupled to a diffusing conserved density. Specifically, we consider a
1+1d random Clifford circuit locally monitored by classically diffusing
particles (``measurers''). The resulting diffusive correlations in the
measurement density are a relevant perturbation to the usual space-time random
MIPT critical point, producing a new universality class for this phase
transition. We find ``Griffiths-like'' effects due to rare space-time regions
where, e.g., the diffusive measurers have a low or high density, but these are
considerably weaker than the Griffiths effects that occur with quenched
randomness that produce rare spatial regions with infinite lifetime.
