Hi,
I tried to check directly on your website again, but currently the search is changed to a white background one.
We tried on your site with the same search theme, but it does not appear: https://youtu.be/w3Va584j3rQ
There is a blink of the input focus for a second, I guess that is some kind of an IOS 15 feature to signal the input, but it’s all right. It is definitely not coming from the search itself.
I assume Apple made yet again something unusual, it is not the first time we are dealing with random issues only affecting some special cases. If you go to this URL, does the red search bar have the same issue on your end: https://ajaxsearchpro.com/isotopic-layout/