Newton's First Law of Motion is that an object in motion will stay in motion unless acted upon by an outside force. Due to the lack of gravity in space, until a different outside force acts upon the object, the baseball will continue to move at a constant speed in the same direction for all time (until acted upon by something else).
It all comes to the doppler effect, the red shift means that the galaxy is moving away from us. The redshift is a result from the doppler effect, so as the galaxy moves away the wavelength expands, increasing the wavelength which responds to the red light.
