This browser (Microsoft Edge) has a zoom feature which I normally have set at 100%. If I change it to 110%, it moves the whole menu off the top of the page and you have to click on the left-top dashed lines to get the menu to come up. Only now it appears down the left side of the screen. The "SEARCH" tag is near the top and clicking it (now it's underlined) brings the search box up underneath the tag in the menu blue area. Moving the cursor to it is not a problem.
So I changed the zoom to 90% to see if that would make a difference. The menu is back at the top of the page but the "SEARCH" tag is not underlined. With that browser setting, the search box is disappearing again. Same as 100% zoom.
So the problem appears to be with this browser when the SEARCH is at the top of the page. I noticed that there is a small sliver of gray background color between the search box and the blue area of the menu. Moving the cursor from the blue area, over that gray sliver, to the search box may be enough to trigger the disappearing act. That might explain why a quicker movement sometimes does not set it off.