The movement of convection currents under the surface, MOVE the tectonic plates and its crust.
Convection currents (as you should know) are formed from nuclear processes in the core of the earth that release great amount of thermal energy, this energy takes the form of a 'convection current' within the semi-solid mantle, which heats part of the mantle, which causes it to expand and rise, much like how convection currents in water work.
When the magma/mantle cools, it begins to fall back down, however, when it reaches the crust, it moves it along in a 'set' direction. It moves the crust just before it falls back down and is replaced with warmer magma/mantle.
This process over time has caused continental drift, and the formation of the landmasses all around the world, and all tectonic processes.
Hope this helps.
-Charlie (*I have copied from another source, but it was my own answer; https://socratic.org/questions/how-are-tectonic-plates-moved-by-convection-currents#521883)
(I have left out the part of Slab Pull being a factor, I will however put that information here if you want it.)
Slab-pull is caused by the cooler, denser crust of usually a oceanic tectonic plate being subducted (going under if you will) the less dense, continental plate.
As it goes down into the mantle and melts, it drags along the rest of the tectonic plate with, though at a very slow rate.
This is how plates are now more commonly believed to move, due to the parts of them cooling and getting denser, and gradually being pushed down and melted, back into magma, in the mantle.
This also helps to explain why the world doesn't expand as new crust is being created (at constructive plate boundaries where it reaches the surface in the gap and cools to form new crust, and), at the same rate as old crust being destroyed (at destructive/convergent plate boundaries where it is melted down).