Re: Reaper users, dragging multiple tracks.
Portions of audio within a track are usually called Regions. It should be possible to lasso the relevant audio regions with the mouse (or trackpad) and drag them together, preserving the timing relationships between them. The automated "snap-to" edit function might help too.
Insert Measure, by definition, adds time between bar lines. If any of your audio overhangs the bar lines, the insert measure edit may divide it.