Since there are some questions about using pastes, I think I can provide some thoughts on the subject. I am very familure with Crox, since I sell both it and hard balsa strops.
Most pastes we use come from some kind of ground up substance and is mixed with a binder to create either a paste, a liquid or a hard bar. Some are oily and some are not. But they all provide nearly the same idea - a product to either polish or grind.
We can go from the most corse products that have carborundum in them (usually black) to those that have no cutting effect at all like jewlers rough (the red) and everything inbetween, white, green, brown, blue, etc.
The problems with past are several. One is their inconsistency in repeat performance. Second is incorrect application to the surface one decides to use it on. Third is their effect to the edge we created. Fourth is over usage.
The inconsistency comes from the fact that pastes are more like a variable stone then anything. I have found the surface goes from bad to good to bad again. But how can this be? The answer lies in the fact that we tend to over apply the stuff to our surface, and until it is worn down some it doesn't work for crap. Then as we use it, it wears off and doesn't work for crap again. Consider putting mayonaise on a sandwich. If we put it on too thick we can move it around all day with the knife and never remove it from the blade without wiping it off on either the bread or the edge of the jar. The same applies with pastes. If it is applied too thick, it doesn't do much of anything except move around and get wiped off the blade. With each successive usage though, it starts to work better. At some point it really works nicely. So first off, we tend to overapply.
So what about the different surfaces?
Well, we have different options here. We can apply our paste to our linen or leather strop, or to a hard leather strop or to hard balsa. But which one is best? Here again, it is up to the individual to choose. Soft linen and soft leather are usually not recommended because of the possibility of rolling the edge. If you are extremely efficient at stropping then this may work for you. Hard balsa and hard leather provide a more forgiving surface when it comes to rolling the edge. With the use of Crox, at least the one I sell, one or two rolls of the edge and you are headed back to the hone. This stuff is extremely efficient. The reason we use Balsa is not because it is cheap, but rather it has the quality of holding the media in its poures so it can be presented to the edge in a reasonably consistent fassion. The surface is soft and we can provide different pressure as we use it. The rough side of leather works somewhat like this also.
So how do we provide a proper application of our paste to the surface we have chosen? I will tell you how I do crox to my balsa. I take my dry crox and put some on my index finger and work it into the surface at several places. Then I take some mineral oil and start to spread it around. I continue to do both until the whole surface is coated. I let this sit for a short period of time and then wipe the surface with a clean paper towel. I continue to do this until there is hardly any crox removed. Then I know the poures are filled and the surface is ready.
How long does this last? My current pasted hard balsa strop has been used on over 160 razors and I have yet to re-apply any crox to the surface.
So how many strokes are we suppost to give to the blade?
Well this is an interesting question. I do NOT use my crox, or any paste for that matter, as a finishing medium. The reason is, as Bart has stated, using pastes puts a convex edge on the blade. I "sometimes" use crox on hard balsa between my 16k and my finishing stone. The reason I do is because I can see a very, very fine wire on the edge and need to remove it before I finish the edge. Ususlly 3 strokes will do that. More than that is to many. When it comes to pastes and sprays, less is better.
When I feel my razor start to lose its smoothness, usually after 6 shaves, I touch it up on my balsa 6 strokes only and then strop on my leather. That usually puts it back to normal for another 6 shaves and keeps me going for 3 to 4 months.
My question to you all is: If you are getting such a smooth and incredibly sharp edge, as you have stated before, why are you suddenly looking for something better? Just asking.