Knee caps move side to side.

rather depends how much they're moving from side too side - some movement is normal, but if they're not doing it in the correct way this could give you pain

runner's knee (I think) is more further down the leg - just below the kneecap

Knee caps are free to move side to side when the muscles are relaxed, this is not runner's knee, do you have pain when you move the knee cap?  is sitting with the knees bent sore?- if so it may be chondromalaciaa patellae- sounds bad, but a few dozen straight leg raises a day will help.

If the pain is at the outside of the knee and comes on after a predictable length of time running, it may be runner's knee- try a cho-pat strap.

Use the internet to self diagnose and then find the thread on here to give you some ideas for treatment.

Spen71, good advice from Tricialitt - the kneecap is a free-floating bone situated in the tendon of your quads (front thigh muscles) and these facilitate its movement so strengthening the quads is a good idea.

Google straight leg raise for info on how to do it. Position yourself so you're crook-lying (half between sitting and lying flat, so supported on elbows). This reduces the effort of the quad as a hip flexor when doing the SLR, targetting the knee more.