You got all sorts of great responses there and I wish you the best of luck with your conditioning of your clients. I would just like to add a couple pieces as well.
My biggest concerns with folks and running have more to do with imposed limitations versus strength deficits. What I mean is that the repetitive and single plane of motion that running presents itself as can be problematic. The average client that we have these days is someone that sits eight to ten hours a day at work, maybe drives another 1 to 2 hours and sits casually another 3 to 4. This creates the commonly observed upper and lower crossed syndromes (i.e. overpronated feet, tight hip flexors, excessive thoracic curvature, etc) From a functional training standpoint, you may want to design programs that are aimed at maintaining proper dorsiflexion, hip extension and thoracic mobility.
Maybe learn a few quick assessments for those joints and being each session with clearing and including those areas as part of the days workout. Try to help identify the clients areas of mobility concerns and create daily homework assignments with the aim of keeping ideal range of motion.
I like the idea of increasing gluteal strength but the only concern that I have is running takes place primarily in the sagittal plane. In most runners this creates tight hip flexors and in turn the IT band gets tugged on pretty hard, thus making gluteal actions inhibited. You probably want to put a person in a single leg stance and watch them squat to see if their knee caves in. This could tell you a whole lot about their lower body function and give you a plan of attack.
In general, if you watch them squat from a single leg and we get a “caving in,” action at the knee, here’s what it could mean (or what you could further look at) from the hip, knee, and ankle:
pelvis gets stuck in anterior tilt, femur interiorly rotates, knee medially displaces, tibia interiorly rotates, foot abducts and flattens out.
With all that being said, I would just make sure to assess by looking and asking questions. If your client wants to run and wants your help to do so, it is your job to help prevent them from getting hurt before anything else.
hope this helps,