Modules disappear on search results page.

The real solution would be to make this info dynamic according the URL of the page where the CP search module resides.

It can be made dynamic. The "our-resources/" part you are adding is derived from Itemid. There are several parameters in Custom Properties that effects Itemid:
- Component configuration:
- Use Itemid: None: do not append Itemid to URL. Current Page: append current page's Itemid.
- mod_cpsearch, mod_cpcloud, mod_cpmenu
- Itemid: force the use of a specific Itemid, thus controlling the landing page of the results
- CPtags plugin:
- Itemid: force the use of a specific Itemid on tags displayed in content element

I suggest you to try configuring your site - in respect of this subject - with SEF turned off, as to see the actual Itemid being appended to the urls.

the URL of the page where the CP search module resides.

The problem is that there can be *many* CP search modules.

This component should be written in such a way that the results pages have the same modules as the search page.

We believe is not responsibility of this component to control module placement. Others migth find such a behaviour too intrusive and limiting. We purposedly used the API's ContentHelperRoute::getArticleRoute to let Joomla return the proper link for an article.

I suggest you to try configuring your site - in respect of this subject - with SEF turned off, as to see the actual Itemid being appended to the urls.

I realize that with SEF turned off the site works fine (see first post).
This further proves my point that the problem is not CP per se, but it's the way Joomla translates Itemid when SEF is turned on. And we only have so much control on that.