and search.action.focusPreviousSearchResult (CTRL+F4)
Original answer
Unfortunately there is currently no command to go to the next entry in the search result pane. As a workaround you can execute workbench.view.search (by default* bound to CTRL+Shift+f) and navigate to the next item via ↓ and select it with ENTER.
When you have executed actions.find (by default bound to Shift+f) in order to find text in the current file only, then you can use the editor.action.nextMatchFindAction command which is usually bound to F3. Its default binding is declared like this:
If you are planning to rebind F4 and Shift-F4 to different keys, please note that there are multiple actions bound to those keys. For consistent behaviour, you may wish to rebind them all.
I know this is an old post, but, it's all I can find on Stack Overflow so, throwing in what I wish I'd found here. If I search (Ctrl+F), just hitting Enter and Shift+Enter take me to the next and previous occurrence of the match. F4 doesn't work for me (on Fedora).