I added some fighting motions for a short scene. When I press play, some motions are being left out. I can only see them all when I put it on By Frame or drag the Transform on the timeline.
Otherwise, it will skip some frames.
What can I do to fix this issue?

Without your system specs it's hard to say. 

You can try previewing in Quick mode rather than High as that will be less taxing on the system.

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Thank you, I will try that

The rendered output is always by frame, so you will never be missing any frames in your rendered movies. What you are seeing is the render not keeping up with the frame rate of the timeline. Just like any game running at lower FPS, you have bigger steps. But there is no data actually missing.

Thanks, I did a test render and it all showed up