I don't think it would be the kit - I regularly use my X-T2 and 35mm f1.4 to shoot my boys and my 18 month old niece. In fact, my 7 year old son got some decent photos of his cousin at the weekend using that set up. It was using eye detect AF, in "S" AF mode, although I'd prefer to use "C", but he's only just got the hang of listening to the focus confirmation beep before fully pressing the shutter.
How have you got the AF set up on your camera? Which of the AF modes is it in? I usually use the "Accelerating/decelerating" setting for toddlers.
You also have to set up the shot somehow, ideally without them realising that is what you are doing, getting them to sit on something usually gives a fraction of a second when they aren't moving:
To the moon! by
Lewis Craik, on Flickr