I haven't seen studies per se, but lithium mining, as with all mining, is a dirty business too. But unlike oil extraction, refining, and transport, the manufacturing of a new battery occurs at most a few times in a car's lifetime, but fossil fuel use is an every week thing.
The discussion I've seen is mostly about coal-powered electrical generation and its impact on the overall pollution from electric vehicles, but given that coal is in such rapid decline in the US the real discussion is gas-turbine generation and transmission costs. (
Note that in Asia coal is still on the rise.). Gas turbine generation is about twice as efficient as coal in sustained output terms, but gas turbines can be stopped and restarted in a very short period of time, while coal plants can't do that. This makes gas powered generation much more responsive overall to the varying loads on the grid.
This is a very interesting study on gas generation done by the Department of Energy:
https://www.energy.gov/sites/prod/files/2016/09/f33/CHP-Gas Turbine.pdf
Given that so much natural gas production is literally a throw-away these days from oil production, and the fact that gas burns so cleanly compared to coal and oil, it looks like a really good idea to use it in place of coal. I also like gas generation versus wind because you can place a gas power plant where the demand is, rather than wind turbines which have to be placed where the wind is, and need long high-voltage transmission lines to get the power to metropolitan areas. Anyone who looks at the process of coal mining probably comes to the conclusion that using natural gas is better for the environment, even if it didn't burn more cleanly, but it does, for a double bonus.
I think the evidence would should that using electricity to power vehicles is probably a lot better than diesel or gasoline for the environment. Distributing gasoline and diesel fuel is like transporting billions of gallons of toxic chemicals every day through cities and the countryside.
Nonetheless, I still think the best long-term solution overall for electrical generation is nuclear.