Recent experiments are discussed where temperature gradients across mesoscopic pores are shown to provide essential mechanisms for autonomous molecular evolution. On the one hand, laminar thermal convection can drive DNA replication as the molecules are continuously cycled between hot and cold regions of a chamber. On the other hand, thermophoresis can accumulate charged biopolymers in similar convection settings. The experiments show that temperature differences analogous to those across porous rocks present a robust nonequilibrium boundary condition to feed the replication and accumulation of evolving molecules. It is speculated that similar nonequilibrium conditions near porous submarine hydrothermal mounds could have triggered the origin of life. In such a scenario, the encapsulation of cells with membranes would be a later development. It is expected that detailed studies of mesoscopic boundary conditions under nonequilibrium conditions will reveal new connecting pieces in the fascinating puzzle of the origins of life.