In a direct current (DC) setup, electrons will be moving continuously though the wires in a big loop. But the displacement of the electrons is not the electrical energy we are interested in.
It is instructive to think about alternating current (AC). In this case, electrons will be moving a bit forward, and then a bit back. So, basically wiggling more or less in place. There is still a lot of energy transferred.
A metaphor is a chain, like in a bicycle. You can use the chain to transfer energy, but it is not the chain links that are the energy. The movement is the energy. This is similar to electricity; it is not the electrons that move that is the electrical energy, instead the electrical energy is the movement itself.
(Due to Einstein's theory of relativity, we know that E=mc^2, which can be summarized as "all forms of energy are also mass". Considering this, all transfer of energy is transfer of mass. But it is miniscule. A small nuclear power plant could produce 4450 GWh in one year. The relativistic mass of this is 180g. All the energy produced in one full year of this plant has a total mass of 180g. So you should not consider electricity to be transfer of mass, even though it technically is :P )
(Also, since you asked for a physicist: I am not a physicist)