Blurring in photos comes from two primary sources - motion blur caused by the subject moving and motion blur caused by the camera moving (ergo handshake).
Both are countered by the same thing, which is using a faster shutter speed. Chances are if you are using those kinds of camera you are most likely using automatic modes built into the cameras. Most default automatic modes tend to assume fairly slow motion, standard hand holding and adequate light (or a subject that can be lit with flash). So they are not pushing for fast shutter speeds suitable for a moving subject.
If you use their "sports mode" that is built into most then the camera will start to aim for a faster shutter speed, it might not get it right every time, but it should improve general results in good lighting.
Cameras can't tell what you are taking photographs of, though face detection and some motion sensors in them to detect their own motions can go a long way to helping out at times. So sometimes you've got to think for the camera. If your Sony has any manual controls you could use them, although manual on compact cameras can be a bit fiddly due to the way you have to access them (if you do for running or sports or action you'd ideally want 1/500sec or faster shutter speeds).
A final factor is also the cameras sensor size and built in editing. A smaller sensor might well hide some edge softeness, especially if the camera is applying a fair bit of sharpening to the photo. Ergo the photo might be "softer" but the smaller size (from the smaller sensor) and the built in sharpening might well hide that softeness.