Connect the black wire of the new 3 wire cable to the common screw of the 3 way switch that you will move to the new location. If you only need two switches leave out both the 4 way switches.
Wiring A 4 Way Switch
Adding a 4 way switch to a 3 way circuit. Hi phil great article helps me understand how 3 and 4 way circuits work. I have an older home with a stair way down to a hallway. The second controls the light in the hall. All you need to do is find the 4 wire that currently runs from one of your 3 way switches to the other. One 3 way circuit controls a light at the top of the stairs. Here is an example of a light in the middle of the run.
You can also add as many 4 way switches as you like in place of any one shown in the circuits that follow. Here are another couple of variations of 3 and 4 way switch circuits. You will need to cutbreak this and add the 4 way switch in between. The 4 way switch will have 2 terminals marked in and 2 marked out. Is it possible to connect 2 separate 3 way circuits to make a single 4 way circuit. All the 4 way switches are optional.
Connect the returning travellers from the new switch box location red and white and connect them to the in or the out terminals on the new 4 way switch. In this application where two 3way switches already exist a 4way switch may be added by simply installing two 2 wire with ground cables of the same wire gauge as the circuit wiring from either of the 3way light switches to the location of the new 4way switch.