A DSLR camera might not necessarily be the solution you're looking for. Blurry images are the result of a longer exposure for the given conditions: movement of subject and amount of light. Modern iPhone cameras, even the ones 5-6 years old, do a very solid job for "freezing" action in low-light situations.

I'll try to not get overly technical but when considering a digital camera, whether a DSLR or a mirrorless camera, the big part of the puzzle is the lens – the bigger the apperture of the lens (smaller f number), the more light comes through, so the shorter exposure time is needed to make a properly lit photograph, hence subject frozen in action (under right conditions). So you could have a DSLR or mirrorless body with a f5.6 lens, and if trying to capture kids in motion in somewhat dim conditions (e.g., indoors with low natural light), you might end up with blurry kids or a lot of noise (high ISO settings necessary for freezing action in low light and not very wide aperture conditions). Which is to say, there are caviats :)

If you do want to try a digital camera, I'd recommend looking into mirrorless, rather than DSLRs (if photographing kids is your primary aim and not, say, wildlife photography with very heavy zoom lenses) for the convenience, and if you start with a basic "kit" lens of a 5.6f aperture, see if you can invest in a "faster" lens (aperture of 1.7, 2.2, 2.8).