Best MILC for Shooting Figure Skating?

I'd not attempt this with a MILC for the simple reason that the AF systems are not designed for sports action of this kind.  The LCD and EVF (viewfinder) on MILCs are also an issue when trying to take action shots - there's a lag between what's happening and what they display, small but a problem for timing shots.

Lenses are an issue.  The range of lenses available for MILCs is not suited to action sports.  On the other hand the cost and weight of serious sports lenses seems unlikely to be attractive to your 16 year old.

There is also the issue of technique.  Is this 16 year old going to develop the technique required for serious action shots of sports ?  I'd be skeptical.

So all in all I'm not sure there's much point in worrying about which MILC or DSLR you get.

Personally I'd prefer a DSLR ( even an entry level DSLR ) to a MILC for these tasks.  Maybe a D5100.  However for a MILC I'd consider a Nikon J1 or V1, simply because the AF system is a little better than average for a MILC.  That said, the lens choices, in the long run, are poor.

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StephenG