The new search field (labels_es) should be filled with data in EntitySchemaContentHandler::getDataForSearchIndex. After this is wired up the search fields should be propagated for all existing entity schemas in beta and production, probably by purging all entity schema pages (this can be done either, preferably, via the API or via the PurgePage maintenance script).
This can be verified using action=cirrusdump.
In our PoC this was done in https://gerrit.wikimedia.org/r/c/mediawiki/extensions/EntitySchema/+/1071938/1/src/MediaWiki/Content/EntitySchemaContentHandler.php.