I am assessing clients at all times. I watch their motion and stability in every exercise they perform.
The first thing that I have all clients “do” is to move through the foundation movements at all of their joints. I caution them to move slowly, stop if they feel pain or that the motion is not a good one for the joint. This takes very little time. Limits in any movement direct further investigation into their ROM/flexibility.
I take this a step further in resistance training. All resistance exercises are performed without resistance/weight first. I don’t ask someone to do a push up if I haven’t seen them move the arms in a manner that will be used for a push up. While 90% or more of clients will have no problem, I have found a number of clients that did. And finding out before they attempt a loading/unloading situation is always better than finding out after the attempt.
And extra attention needs to given to any issue that has been discovered during the course of training/assessments. If a joint or muscle/muscle group has had issues/injuries in the past, any movement involving that area should be done with that in mind.
This answer is not nearly as indepth as I would be with a student or client. There are many variables to every movement and every client. Assessment is ongoing, any exercise or movement can provide feedback on whether a client needs regression, reassessment, or more instruction.