It's due to the fixed menu position, where the search is placed. It always stays on the top, regardless of the page scroll, and since the search is positioned in there, it has the same behavior.
One solution would be to disable the menu fixation on smaller screen devices. I would probably recommend this, as it gives more visibility of your site.
Another is to perhaps use two search instances, one for bigger screen sizes, another for smaller ones in different positions. The desktop version would stay where it is now, but the mobile version would be in a different position.Best,
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