  We investigate the phase transitions of the $q$-state Brownian Potts model in
two dimensions (2d) comprising Potts spins that diffuse like Brownian particles
and interact ferromagnetically with other spins within a fixed distance. With
extensive Monte Carlo simulations we find a continuous phase transition from a
paramagnetic to a ferromagnetic phase even for $q>4$. This is in sharp contrast
to the existence of a discontinuous phase transition in the equilibrium
$q$-state Potts model in 2d with $q>4$. We present detailed numerical evidence
for a continuous phase transition and argue that diffusion generated dynamical
positional disorder suppresses phase coexistence leading to a continuous
transition.
