An expression for "Lying on your stomach, kicking your legs and feet up in the air"

Lying on your stomach, facing down is called the prone position. Instead, supine is lying straight on your back.

 

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The image above shows a woman performing a prone leg lift

 

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If the leg is not bent at the knee, it is called prone straight leg raise / lift

 

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When both legs are raised simultaneously it's called frog leg lifts
Source

 

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And finally, if the person is kicking their legs, it is called prone flutter kicks

YouTube video