It is NOT directly activated by crunches, situps, or other typical "abs" workouts. It is mostly activated subconsciously (before other abdominal muscles!), and in many people that is enough to keep it strong. However, in people who sit or lay a lot, such as people who have desk jobs or disabilities or health conditions that prevent them from standing or walking much throughout the day, the automatic activation can "turn off"! Then you get a weak transverse abdominis, even if you do abs workouts, unless you consciously activate that muscle.
I've been doing weight-training several times a week for about 21 months now. A few months after I began, I injured my back by continuing an exercise while I was feeling discomfort (it was one I had done before without incident), and I didn't understand why it happened. It took 6 weeks to heal, and I couldn't work out at all for 3 weeks because every exercise I would have done caused it to twinge and threaten that horrible pain. I now know this is because your transverse abdominis is automatically used anytime you lift an arm or leg, and I am pretty sure that is what I injured.
There are only two movements that have been found in studies to focus on strengthening the transverse abdominis: drawing-in and bracing. Drawing-in is where you pull your belly-button (navel) towards your spine. Bracing is where you tense your belly as if someone is about to punch it. Some studies show one is better than the other, but there isn't a lot of agreement about which is which. Personally there are some exercises where I can't do one or the other because it feels like rubbing head and patting belly, so both are useful for different things.
After I learned about this muscle I began tightening it every time I use a weight-training machine, and I have not had an injury since.
Last year I also began to lose fat (not weight), which made my belly sway more and lean out from my body more, because it was less dense and thus didn't stay in place. (Similar to how a very full backpack will stay in place as you walk, but a half-full one will sway from side to side) I wondered if this was creating strain on my back, but of course there is no information out there. Because no one studies the practical aspects of fat people working out, just the irrelevant and useless effects like "do you get lighter" or "do you get narrower."
I decided to believe in my own experience and give my back a "rest" after a workout (or a long walk!) by wearing a brace that wraps around my abdomen for about 30 minutes after. Since I started doing this I have had no back pain. Even more telling, I haven't had that weakness feeling where I feel like my spine might snap in half, like I used to sometimes after a workout. Once I learned about the transverse abdominis this made a lot of sense-- even before I tensed it on purpose, it was getting worn out supporting my spine as I exercised my arms and legs.
So if your back feels weak or gets achy regularly after a workout, I recommend trying a back brace to rest this muscle after a workout. And YES, losing fat can make your back hurt.
When I stopped working out 3 weeks ago because the gym is closed, my back started feeling weak again. So last week I started doing exercises for my transverse abdominis a few times throughout the day, and it is already helping. I do "bracing" while laying on my back or side and doing movements with my legs. One that I do is holding my legs out straight, bringing them up to vertical and lowering back, and another is holding my legs up vertical and swinging them side to side. With these I have to press my hand to my belly to feel the tension or else I will forget to "brace" which is the whole point.
So now, for any person I know who begins working out, especially for fat people whose belly sways or droops like mine, I would definitely suggest training yourself to tense up your belly as you do each rep (as long as this doesn't hurt). Initially I had to place a hand on my belly to give myself a physical reminder to brace, but now I do it almost without thinking. I also noticed my belly getting tired about halfway through the workday as I just sat at work, so I think it is actually starting to be automatically activated again.