Moss is not a fast plant, and knowing that before you start saves you from both giving up too soon and expecting a carpet by next month. Here is what to actually expect, and why the honest answer is a range rather than a single number.
Why there is no single figure
People want a tidy answer, so many millimetres a month, but moss will not oblige. The rate depends on the species, the moisture, the light, the surface and the temperature all at once, and change any one of those and the growth changes with it. A feather moss racing over damp shaded soil and a slow cushion moss inching across dry stone are both simply "moss", yet they behave nothing alike. For many species the fair measure is a few millimetres of new growth in a whole year, and some of the slow cushion and mat mosses barely manage even that. Treat any confident single number you read with a pinch of salt.
Realistic timescales
Kept constantly damp and shaded, a slurry or a fresh transplant usually starts to green up and grip within a few weeks. A patch knits into continuous cover over a few months. A convincing moss lawn or wall takes a full season or two to look established, and the deep, springy cushions of an old moss garden are the work of years, sometimes many of them. None of it is quick, but most of it is steady, and it rewards you for leaving it alone.
What speeds it up
Three things above all: constant moisture, deep shade, and firm contact with the surface. It helps to remember that moss has no roots, so it cannot draw water up from below; its thread-like rhizoids only anchor it, and it takes in water and the little it needs across its whole surface. That is why keeping the surface itself damp matters far more than watering the ground beneath it. Mild temperatures help too, which is why moss does much of its visible growing in the cool, damp shoulders of the year, spring and autumn, rather than high summer. Soft rainwater, a humid spot and shelter from drying wind all push it along.
Why it seems to stop
Moss is what botanists call poikilohydric: its water content simply tracks its surroundings, with no roots or waxy skin to hold moisture in. When the air dries, the moss dries with it, shrivelling and going dormant until the next wetting, so a patch that looks dead or static in a dry spell is usually just waiting. It has not failed; it is paused. Wet it again and it resumes, often turning from grey and crisp back to green within minutes. The single biggest mistake is deciding it has died and giving up during a dry fortnight. Patience is most of the technique; see the growing guide.
Judging progress for yourself
The useful thing is not a figure from a website but the habit of watching your own patch. Mark the edge of a transplant with a pebble or a photograph, then check it after a month, and again after three. In good conditions you should see the gaps closing and the colour deepening; if nothing is happening, the cause is nearly always that it is drying out between waterings, not that the moss is idle. Moss grown from a slurry can look like nothing at all for weeks while the fine early threads establish, then seem to arrive all at once. Judge it by the season too, since a patch that sat still through a hot dry summer may well surge the moment autumn rain returns.