Magnetic skyrmion is a swirling topological spin texture behaving as an individual particle. It shows a gyro-motion similarly to that of a charged particle under a magnetic field, being led to the transverse shift to the electric current, i.e., skyrmion Hall effect. With the open boundaries of a sample, this results in an accumulation of skyrmions on one side and their depletion on the other side. Here we demonstrate experimentally that this effect propagates non-locally over tens of micrometers even where the electric current is absent, when the narrow wires bridge bar-shaped Pt/Co/Ir heterostructure thin film systems. This nonlocality can be understood in terms of the "chemical potential" gradient for the skyrmion bubble induced by the skyrmion Hall effect in the nonequilibrium steady state under the electric current. The present result shows that the skyrmion Hall effect acts as the skyrmion pump and the thermodynamic concepts can be applied to the aggregate of skyrmion bubbles.