  This paper is about the dynamics of non-diffusive temperature fronts evolving
by the incompressible viscous Boussinesq system in $\mathbb{R}^3$. We provide
local in time existence results for initial data of arbitrary size.
Furthermore, we show global in time propagation of regularity for small initial
data in critical spaces. The developed techniques allow to consider general
fronts where the temperature is piecewise H\"older (not necessarily constant),
which preserve their structure together with the regularity of the evolving
interface.
