How has river to ocean flux changed over the last 500 million years?
There is no direct answer. One way to think about this is to consider the water cycle. Water evaporates from the oceans and is precipitates over land. Distance from the ocean affects the amount of precipitation so at the center of a large contintent you may have less precipitation than over a smaller one. Mountains also drain moisture out of the air as it is heaved. hence the topography of the continents affect this effect. A Pangea directly after collisions will likely have more and higher mountains than a later stage when break-up is imminent. Location on the earth's surface is a third factor since we have larger scale circulation belts that affect the atmospheric conditions so that we have a tropical belt (the Intertropical Convergence Zone) around the equator, a drier zone north and south of the ITCZ, followed by temperate and polar zones towards the poles. Hence the location of the continent(s) and their topography will affect the zones and the zones will will determine which parts receive more or less precipitation. Furthermore the movement of the continent(s) through these climate zones change conditions as well.

In summary, we can be sure it is not constant over geological times. The only possible way to get some relative handle on the question would be to try to apply a global circulation model to past continent configurations, including topographical effects and see what comes out, being aware of all uncertainties and comparing with geological evidence.