Roof Moss Removal from Concrete Tiles
First off, it's important to know what type of roof that you have. I'm focusing here on concrete tile roofs.
I have found a pesticide to kill moss on concrete tiles. It is same as used on composition roofs, but it is approved for concrete tiles by pesticide manufacturer. It also has the preventative properties of zinc used on composition roofs.
The most effective way of removing clumpy moss is to brush it off on a dry, warm day. After the moss has been removed, the roof can be treated to kill the spores and prevent new growth.
About Concrete Tiles
Sometimes called "slate" (but not a true slate), concrete tile is a light-weight, porous concrete that is shaped to fit over slats of wood nailed to your plywood roof. The concrete tiles interlock with each other. Most of them will probably not be nailed down, but are held in place by the interlocking parts and by the weight of the overlapping tile above and to the side of it. It's a pretty neat system (for dry climates).
That being said, I think it's obvious also that concrete tiles are not waterproof. They are not meant to hold water, they are meant to shed water. This is why it's important to keep concrete tile roofs clean of debris, especially in valleys where the water naturally collects as it flows downwards. Clogged valleys block water flow from rain. The water backs up and overflows the lips of the metal valleys and goes into the insulation and ceilings below. This is a big nuisance that may require sheet-rock repair and painting. In bad cases of dirty valleys, the tiles can be temporarily removed and the valleys cleaned out.
Since concrete tiles are on the roof, the roof structure must be built to withstand the weight of the tiles. This is one reason why the tiles are so porous. The manufacture wants to make them light for the roof so the concrete has lots of air in it. This is good and bad. Good because it's easier to transport, handle and support on the roof, but bad because tile is so porous and holds water for moss after the finish has worn off.
When new, concrete tile has a finish or glaze on the top surface. This glaze repels water. As the ultraviolet (UV) light from the sun strikes the tiles, it breaks down the glazing and causes it to spider or micro-fracture. The water is then able to penetrate the tile. As the tile soaks up water and allows it to pass through, it leaches the cement from the tile and makes it more brittle. This water wicking through the tile also slow feeds moss spores which hang out on the bottom edges of the tile and in the side interlocks.
Pressure washing removes much of the finish from the concrete tile roofs and gets mud (dirt and water) to get blown behind the lath pieces that the tiles rest on. This provides a place for the moss spores to hang out on. If you pressure wash the tile, it may require resealing in order to not void the roof warranty.
Replacing Broken Tiles
Maintenance of broken tiles is done by lifting the tiles on the row above and lifting out the broken tile. Then a new piece can be inserted to replace the broken piece. The new tile will often vary in appearance/finish from the older tile. Sometimes for appearance an older tile from a less-seen part of the roof can be used to replace the broken one. Then the newer tile can be put in the less-seen part of the roof. I recommend that you keep about 5 tiles per year of expected future life of your roof. For example, if you have a 50 year warranty, and it's been 7 years, then I recommend you keep (50-7 = 43 years) 43 years x 5 tiles per year = 215 tiles for replacement. The reason I recommend keeping tiles on hand is that the manufactures discontinue styles and you may not be able to get any of the same fit/color tiles when you need them later. You can remove a piece of tile from your roof (temporarily) and take it to roofing store that sells them to get exact replacement. The manufacturer is often shown on the bottom because it was part of the mold that was used to form the tile.