The recently discovered spin Seebeck effect refers to a spin current induced by a temperature gradient in a ferromagnetic material. It combines spin degrees of freedom with caloric properties, opening the door for the invention of new, spin caloritronic devices. Using spin model simulations as well as an innovative, multiscale micromagnetic framework we show that magnonic spin currents caused by temperature gradients lead to spin transfer torque effects, which can drag a domain wall in a ferromagnetic nanostructure towards the hotter part of the wire. This effect opens new perspectives for the control and manipulation of domain structures.