This should be automatic... If I understand correctly.
If lets say I have a project with 6 video clips in sequence on 1 track, with the audio for each on the second track, I then place the timeline on the third video clip, and halfway through this clip, I decide to edit out something from within the middle.
I place the timeline, then make my first cut, I think position it again for the other cut, the material between the two cuts is the material I want to delete. so I can click on it, and it highlights (both video and audio) then I can do one of the following:
1. I can use the button combo Ctrl + Del, which will delete the material and move everything right of the material left to fill the gap (this includes material that is not on the same track) ... I think this is what you want.
There is also a feature in the cut icon area that you allows you to remove scene, you would select the material so it is highlighted, use the down arrow where the cut/split scene icon is, and choose Remove Scene. this will do the same thing as (1), but the icon for your split scene is not set to "Remove Scene", which is why I prefer the Ctrl + Del function, it is a shortcut key, and it keeps my cut/split icon where I need it so I do not make any unnecessary removals.
2. Delete the material then group all other objects (left click objects then use the CTRL + Left Click to select additional objects) then group them... once they are grouped move the last part of the third video to the left to fill the void.