Currently, I have a 6' x 6' compost heap 4 feet high layered up from last year's compost, the Maple leaves and Mulberry leaves from our back yard, garden green waste cleanup, etc. Using a pitch fork, I slit into the top of the pile down a foot to make a trench that the kitchen bucket goes into.
This time of year, with the cold, the compost gets hot enough to gently steam, but the worms move out of the warm areas to consume leaf matter and kitchen waste. As long as there's warmth, they have the mobility to take care of themselves.
I've turned the pile twice since September and have a quite heavy worm population consisting of angle worms (yellow banded), standard earth worms and night crawlers.
The Maple leaves have quite a bit of sugar which the bacteria likes (heat), the Mulberry leaves promote quite a bit of fungal growth, you can see the mycelium strands. Night crawlers eat leaves, bore into Kale and Cabbage stalks and do a number on anything they can ingest.
Basically the amount of heat depends on the leaves and the grass content. Something like Sycamore creates a pretty cold pile that needs quite a bit of fungal growth to break down unless you shred them and layer in greens. In the spring, this can produce a pretty hot pile that would be a worm killer with 2 inch leaf to 3 inch grass layers.
The coldest pile I ever put together was Oak and Sycamore stacked about 4 feet high. The leaves got wet and compressed so tightly together that it resembled the "Shoe Event Horizon" per one of the Hitchhiker's Guide skits. Trying to turn it did nothing. After two years, I had to sharpen a shovel and chop into it to break up the leaf matter so it would start to break down at which point it went into the main pile as a grass mix and got so hot it cooked everything out. Not worm friendly.
Basic decomposition process: Green or high color sugar bearing with protein, eaten by bacteria with lots of heat. Brown, mostly cellulose and lignins, eaten at a cooler temperature by fungi. Once the rot sets in, worms will eat everything that still has food content.
So, depending on managing the content, yes you can toss it all in the same pile and have a broad range of stuff the worms will eat. They're omnivorous.