  We analyze a family of degenerate parabolic equations with linear growth
Lagrangian having the form $u_t=\div (\varphi(u)\psi(\nabla u/u))$. Here
$|\psi|\le 1$ and saturates at infinity. We present a simple and natural set of
assumptions on the functions $\psi,\varphi$, under which: 1) these equations
fall in the framework provided by \cite{ACMEllipticFLDE, ACMMRelat} and hence
they are well posed, 2) we can ensure finite propagation speed for these
models, 3) a Rankine--Hugoniot analysis on traveling fronts is also performed.
On the particular case of $\varphi(u)=u$ we get more detailed information on
the spreading rate of compactly supported solutions and some interesting
connections with optimal mass transportation theory.
