Ensure the relevant video and audio tracks are armed and double click on the clip after where you want to insert the new video clip to highlight it, then hit shift and spacebar together. This moves that clip and all video after it to the right by the length of that clip and then you can move all the video to the right as far as you like by extending the silence. Pop the new clip into the silence and drag the end of the silence back to remove the gap. All should be in sync.

There is probably a better way that hopefully someone else will suggest, but one way would be to switch from the default of "Insert" editing mode to "Drag and Drop" editing.

   https://cinelerra-gg.org/download/CinelerraGG_Manual/Drag_Drop_Editing.html

Then create groups so that when you insert something, all of the tracks in the group alongside will move as one.

   https://cinelerra-gg.org/download/CinelerraGG_Manual/Grouping_edits.html

I've built a meeting template where we would edit a date in a title, then drop a video in between the intro and the closing sections.  There's 1 video and 2 audio tracks.  How do I drop a video into the middle and have the closing video and audio on the right stay in synch?  The audio doesn't move and stay in sync.  I haven't found it in the docs yet.

Thanks