It's a bit hard to answer the "dirtiest fuel you can find" case specifically, because there are a number of areas where EVs are more efficient. The biggest difference is probably the fact that the internal combustion engine in a car is about 25% efficient, though it depends on the RPM it is running at etc (the reason hybrids can push it up to ~40-ish% depending on how new they are is because the engine is always running at the most efficient RPM when it runs), but the "dirtiest power" specifically would probably be a coal plant which is only about 30% to 40% efficient (9000 to 11000 BTU/kWh in imperial units) due to low temperatures and the inability to run it as a 2-stage, combined-cycle sort of setup (modern combined-cycle gas turbines are ~60% efficient which is one contributing factor to gas-based electricity being cheaper even though gas costs more per GJ, though since the price of natural gas is quite volatile that sometimes changes). Of course the transportation is different depending on which fuel is used for a power plant.

In the worst-case scenario, accounting for the ~90% efficiency of the electric motors... Well, Xunmin et al. (2005) estimates 3–36%, so lifecycle emissions could be reduced by as little as 3% if you power it 100% by coal, which would be less than the what you'd get from a hybrid, but... You're not really going to find a power grid that is powered 100% by coal these days, even in China. Really the biggest advantage of a BEV, and any other electrification, is that if there are future investments in the grid (and there will be since generators don't last forever) you don't have to replace the engine of your car for it to automatically reduce emissions. The efficiency gains are just a cherry on top.

[Xunmin]: https://www.sciencedirect.com/science/article/abs/pii/S17505...