For action shooting, the Z3 will be the best of the bunch.

Here are the timings for the cameras you mentioned.

The first number is the time (in seconds) to capture the photo if you pre-focus by holding the shutter button half way down to achieve focus lock.

The second number is the time (in seconds) to capture the photo if you do not pre-focus.

Z3 - .1 - .2
Z2 - .1 - .4
S5100 - .1 - .6
770 - .2 - .8

Most of these cameras also have a "burst" mode that let you take photos in fast succession. This is a very handy feature.

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