Hi!
Thank you for the details, I have checked your site and found the issue.
The search box container no longer has a z-index value, to prevent issues like these. I’ve looked through the parent elements, where the plugin is placed, to see if perhaps any of them has any z-index set.
The immediate element by the navigation, where the plugin is placed has a z-index value of 9999, that is causing the problem. (the related CSS rule can be found in the style.css file on lines 841-847 in the child theme directory, according to my developer tools console)
The simplest solution is to override it with a custom CSS, like so:
[html].bp-search-home {
z-index: 1 !important;
}[/html]
That’s it! This should solve the issue.