Unfortunately nobody was around to measure the velocity of tectonic plates 6,000 years ago or 6 million years ago, but we do know that the velocity of plate movement has stayed relatively constant over time. Let me explain one of the ways we know.
Consider the Hawaiian island chain. The Hawaiian islands were formed by volcanic eruptions. There is a volcanic hotspot that oozes magma onto the Pacific Plate. Relative to the hotspot, the Pacific Plate moves NW at 7 cm per year. If plate motion has stayed relatively constant over the last few million years, we would expect to see a new island directly over the hotspot, and the islands would get older and older the further NW of the hotspot they are. Let’s visualize it in a graph. Let’s create a graph that has the age of the Hawaiian islands on one axis, and distance from the Hawaiian hotspot on the other axis. If the plates have been moving at roughly the same velocity since the hotspot started erupting, we would expect to see a linear relationship on our graph. If the plate velocity is variable (as UM suggests) then the relationship would look very irregular and scattered.
I am happy to report that there is a wonderful linear relationship between age of the Hawaiian Islands and distance from the hotspot, suggesting that the Pacific Plate has been moving at 7 cm/year for many millions of years. Take a look at this link from the Hawaii Center for Volcanology and see for yourself.