For the sake of exercising the transverse abdominal muscle it helps to think of this transverse abdominis as having three parts.
In general the transverse abdominis could be thought of as a belt like (or cumberbund-like) muscle that runs horizontally around the waist.
It's function (in part) is to help stabilize the sacro-iliac joint, lumbo-sacral joint and lumbo-thoracic joint.
The upper most band of the transverse abdominis attaches to the inside surface of the lower ribs with fibers interdigitating (interlocking) with those of the respiratory diaphragm.
At the back of the ribcage this band of the transverse abdominis may be opposed (or complemented) by the serratus posterior inferior muscle which attaches from the lowest four ribs to the lowest two thoracic vertebrae and upper 1 or 2 lumbar vertebrae.
The middle band of the transverse abdominis fills the space between the ribcage and the pelvis. It attaches, via the thoraco-lumbar fascia to the lumbar vertebrae.
This portion of the transverse abdominals passes behind the quadratus femoris, which attaches from the pelvis to the lumbar vertebrae and the lowest (12th) pair of ribs. This band of the transverse abdominals may add tension to the quadratus lumborum when engaged.
The lowermost band of the transverse attaches to the front of the pelvis at the ASICs (the front "points" of the hip crest) and to the inguinal ligaments, visible as the lines that separate our thighs from our bellies.
Since the pelvis potentially hinges at the si joints and the pubic bone, this portion of the transverse abdominals may in part oppose the pelvic floor muscles, but also may act to add tension to the thoracolumbar composite, the mass of fused connective tissue at the back of the sacrum that helps to hold the sacrum, lower lumbar vertebrae and hip bones together.
Via the TLC the lower band of the transverse abdominis may have an affect on the sacral multifidus. In general the lower transverse abdominals may have a role in controlling sacroiliac joint stability.
All together the three bands of the transverse abdominals can have an affect on sacroiliac, sacro-lumbar, lumbar and lumbo-thoracic stability.
So then, how do you exercise the transverse abdominals?
The upper and lower layers attach to the fronts of the ribs and pelvis respectively.
Once you've learned to activate each of these three bands a simple transverse abdominis exercise is to activate them sequentially, lower, middle upper, or upper, middle lower. You may find that activation of these naturally causes an exhale. Relaxing them then should allow an inhale.
A variation of this transverse abdominis exercise is to activate all three bands, keep them active and then inhale. Relax them as you exhale.
Note that the feeling of activating the rectus abdominis is more like trying to squeeze the sternum towards the pubic bone while resisting them moving towards each other at the same time.
Transverse abdominals activation should feel different.
Since the transverse abdominis attaches to and crosses behind muscles that attach directly to the lumbar spine and sacrum you may also be able to exercise the transverse abdominals by trying to make the lumbar spine (and sacrum) feel straight and strong.
It's similar to the feeling of giving someone the middle finger but with an extra solid middle finger. Try to create that same feeling of tension when straightening the lumbar spine. Try to create that same feeling in the sacrum also.
A method for indirectly activating the lower band of the transverse abdominis is to engage the pelvic floor muscles. This could be thought of as mula bandha and uddayana bandha.
Engaging the pelvic floor muscles, including:
causes the ischial tuberosities to draw inwards. The lower transverse abdominals automatically resists this inwards pull by pulling inwards on the ASICs and inguinal ligament. Thus the sacroiliac joint, lumbo-sacral joint and pelvis as a whole is stabilized. This action then may go well with making the sacrum and lumbar spine as a unit feel long and strong.
Generally any long term holding of any muscle is bad. Muscles are designed to activate and relax. And so to that end rather than trying to keep transverse abdominals activated to practice activating it and relaxing it when exercising it.
In particular it may be helpful to practice activating it in the context of other exercises, say lower back strengthening exercises.
In terms of yoga poses, it may be helpful to activate the transverse abdominals in twisting exercises. Or if you like, think of twisting as a potential exercise for the transverse abdominis.
Also any poses where the legs are supporting the body but the torso is bend sideways or reaching sideways without the support of the arms.
Transverse abdominus activation can also be handy in standing exercises for low back pain.