How does discontinuous distribution provide evidence for evolution?
There are probably several relevant perspectives. Since you mentioned continental drift, consider a taxon (i.e. species, family, etc.) that evolved when all the continents were joined together to form the supercontinent Pangaea.

If it was a successful species that continued evolving and expanding its range, then we might expect to find its fossils - or maybe even living descendants - on multiple modern continents. In fact, dinosaurs evolved on Pangaea, and their fossils are found on every modern continent.

On the other hand, imagine a taxon that evolved after Pangaea broke up into Laurasia and Gondwanaland. If this taxon evolved in Gondwanaland, then its fossils and descendants will likely occur only on the southern continents. (Gondwanaland broke up into Africa, South America, Australia and Antarctica.)

In fact, you mentioned a good example - lungfish. On closer inspection, lungfish fossils have been discovered in North America, however. So we might guess that lungfish evolved in Pangaea but died out in the northern portion (Laurasia) for some reason.

Elephants belong to the order Proboscidea, and proboscideans were once widespread in Eurasia and North America, and they even ranged into South America for a time. The familiar mammoths and mastodons were apparently exterminated by changing climate and/or human predation. This tells us more about extinction than evolution.

But the fact that proboscidean fossils are found only in the Northern Hemisphere and South America tell us that they evolved in the Northern Hemisphere, spreading into South America only after it was connected to North America.

I'm not sure when proboscideans first evolved, but it was probably long after Pangaea broke up. However, North America and Eurasia have connected and reconnected at various times, making it possible for species to migrate between the two continents. The most famous link is the Bering Land Bridge, which existed during the last "ice age" (Pleistocene).

At any rate, elephants today survive only in the tropics, which, like the sea, are a sort of refuge for "living fossils."

There are a number of taxonomic groups - living and extinct - that are found only in the Northern Hemisphere, only in the Southern Hemisphere, or only on particular continents that are evidence of both evolution and changing environments (including climate change and continental drift).