  We study pluripotential complex Monge-Amp\`ere flows in big cohomology
classes on compact K{\"a}hler manifolds. We use the Perron method, considering
pluripotential subsolutions to the Cauchy problem. We prove that, under natural
assumptions on the data, the upper envelope of all subsolutions is continuous
in space and semi-concave in time, and provides a unique pluripotential
solution with such regularity. We apply this theory to study pluripotential
K{\"a}hler-Ricci flows on compact K{\"a}hler manifolds of general type as well
as on stable varieties.
