Then do you have any recommendation for cameras which can take fast moving objects (not DSLR)?
No camera is perfect expecially Mirorrless. I use own Olympus EM5 OMD, Olympus EPL-5 and some good lens before switching to X100s, X-E1 and X-Pro1 with various lens. Here are my own opinion and own experience with these cameras
OMD, EPL-5
Olympus focusing is very fast, but somehow feel that there is lag time between the focus locked and when the shutter actually capture the image. I tend not to get the focus point I aimed when taking ‘moving’ children photos. Image quality is great but the skin tone is not to my liking. I am not blaming the camera as I see people taking great photos with the same settingsm, possible that the user is at fault. I sold off both Cameras.
X100s
Few months later bought X100s. I was happy with everything that X100s provide and quite close to what you see on the online reviews. What I like most is small, total silence shutter, great skin tone and image quality in good light. Focusing although not as fast as Olympus, but when the focus is locked on target you hardly get blur images. Only flaws I don’t like about the camera is the artificial skin tone at low f-stop for indoor/night shots. The camera is also capable of good studio shots as compared to Olympus. After a while I feel that something is missing, the focus range is too rigid as I shoot anything and everything, my baby, travel, landscape, macro. Guess I’m not a 35mm fan and decided to try X-Pro1.
X-Pro1/X-E1First I bought X-Pro1 with 35mm and absolutely fall in love the image quality, DOF and camera control. Focusing with the lens isn’t very fast but the image it produce is very rewarding. It is very close to the quality of my D800 with Sigma 50mm lens. I decided then to trade my X100s off for X-E1 with 18-55mm lens. What I can say is that the camera has the same sensor and focusing mechanism as X-Pro1 and with XF18-55mm, the focusing is spot on. In my opinion its great combination for taking children photos. I sold the camera and lens to my friend to encourage him to take up photography at a good price. I have been using x-pro1 with XF35mm, XF14mm, xf60mm, XF55-200mm and could get good results too.
Please note that above is based on my limited knowledge, preference and experience. I believe other APSC, Full frame mirorrless camera are as capable also. My opinion is that if you have intention to change your camera, borrow from your friend/rent from shop to try it out first before buying. Else you will regret if they do not perform to your expectation.