End of the Line

You need to take over control of the cart, after the "capture circle" surrounding the carts is overlapping with the capture point.

It does not have to be controlled by the other faction; capturing it from neutral counts as well.

So far i find that this is the easiest way to do this ach:
During prep phase before each game starts: all of the 3 mine carts are moving by themselves to the depot. Just before each game starts, there would usually be 1 or 2 carts close to capping. All you have to do is run and "tag" the mine cart from grey when it is almost reaching the depot and then capturing it.

Depending on your faction some mine carts will be closer to your faction's spawn than opposing so you will definitely reach it before the opposing team does. For alliance it's the NW cart and for horde it's the one headed SE.