I've noticed that when I save a project or export a layout if I switch windows while the project is saving or exporting that ArcGIS PRO will revert to the window where I was saving/exporting when the operation is complete even if I'm actively working in another window.
I would like it of Saving/Exporting was kept in the background if I'm working in another window.
If you'd like to test this try opening a Word document, then go save your ArcGIS PRO project. While the project is saving switch back to your Word document and type. When ArcGIS PRO finishes saving it will automatically interrupt your typing in Word to show you it has finished saving the project. You can also try writing an email or anything else, ArcGIS PRO interrupts anything else I work on while it saves.