Pioneer 1 is a spacecraft launched in 1977 and is now about 11,000,000,000 miles from the Sun. and travelling at 300 million miles per year. The gravity from the Sun is tiny at that distance so I guess you could say that Pioneer is in motion (relative to Earth) and (almost) in equilbrium
A car driving along a straight highway at a constant velocity is in motion and equilibrium.
The force of the engine moving the car is equal to the force of friction on the tires and the axles plus the losses in the transmission and engine when moving at a constant velocity. Cruise control is amazing.
If the driver accelerates, brakes, or turns, the equilibrium is lost, at least momentarily, until steady conditions return.
A plane in level flight or a boat moving on a calm lake are other examples. In these cases, equilibrium is achieved by adjusting the heading to compensate for movement of the supporting fluid.