Copper grease is used for an anti-seize lube, mainly for things to be tightened and left like that, until undone (like a bolt, so it doesn't seize), but as the slider is a moving part, and it doesn't have grease water repellent properties, it's not ideal for the job.
Something like LM grease, it has better water repellent properties, it can be used in bearings (moving parts) and also can withstand high temperatures (like brakes provide), so is perfect for the job.
Alot of people use "red grease" i've not used it myself, as I have some Castrol LM, and has been perfect since
(also alot of people say not to use red grease too, as it dries out)
PS just to add, the person before me used copper grease, it basically dried out from the movement, gathered on the end, damaged the rubber boot and allowed crap/water in, and was seized good and proper to the carrier