Compost requires moisture and turning on a regular basis for best results.
The coarser the ingredients, the longer they take to break down. Branches
can take what seems to be a lifetime, but that's what happens in the woods,
and the soil out there has a beautiful odor. When we harvested firewood in
the forested area, we stacked branches and other trimmings for the wildlife
habitats which eventually rotted. I met a gardener about thirty years ago
who stacked all his broken branches into piles that were left to break down
naturally, and composted other stuff in the conventional manner without
inclusion of large branches because it was easier for him to handle.
I have a chipper/shredder that I use for reducing branches and other coarse
material to compostable size, but I'm not inclined to spend money on city
water when weather will (eventually) provide moisture. I have several wire
bins for composting and, like you, turn the heaps occasionally. The compost
is usually ready for the next garden season, if not sooner. Even if left
alone, it'll just take a little longer.
So far as the occasional fallen limbs that you describe, they earn their
keep by opening small areas for various organisms to breathe while
converting the compost ingreadients. . . it must be hard work for those
little guys, and they need to catch their breath every once in awhile, no?