First you do an evaluation, body movements and strength tests.
If you find that your client is really lacking in core strength or function then you could do a general warmup on a treadmill or something the do your core work ( and don’t forget that the glutes are the core stabilizers of the posterior lower part of your core.)
An example when I think they should be done first. Lets say your client has a posterior pelvic tilt, or low back arches while doing a squat. This can be due to weak abs. This is when abs will be a priority and then you should do them first so that the client is fresh and hasn’t worked out so they won’t be tired.
Also anyone over 40 or so really can’t activate their Transverse ab, i.e. draw in maneuver, bring our belly button to your spine, not being able to bring a blood pressure cuff from 40 ml to 70 ml of pressure when placed in the small of your back etc. So putting these exercises would make sense. Because they can’t even activate properly the most important muscle of all.
Here is an example of when to do them first for the back side of the core. If someone is hamstring dominant while doing a bridge. Which means that there hamstrings work more than the glutes then I would put glutes as a priority first, after you foam rolled then stretched the hamstrings, then do a bridge for the back side of the core…
The point is that you want to figure what your client needs most then put it first or on a day all by itself if you think it will negatively affect the rest of the workout because their abs are tired.
And if you know how to check the strength of the right verse left side of the internal and external obligues and you find one side weaker then the other then doing the weaker side first may be a wise thing to do as well.
All things considered and equal, it really doesn’t matter, but you could also ask your client “We could do abs first if you want, it really doesn’t matter. But if you want to focus on them then we could do them first when you have the most energy?” If they say “yes” then your client will be happy and feel special that you got their opinion on that doesn’t really matter either way. This is PERSONAL training!
I would go based on what you have planned for your client’s session that day. For example, I tend to do a lot of plank work with my clients, and occasionally their arms will be too fatigued by the end of the session to stabilize well in a plank position. So, in some cases I may start a client out with abdominal work.
As long as your client is warmed up properly, you can mix it up and include it at the beginning or at the end.