Mosses (bryophytes) lack vascular tissue (i.e., xylem and phloem) to transport water and other materials and must absorb all their water and nutrients directly from the surface they are living upon or from the surrounding atmosphere and pass them from cell to cell. Therefore, mosses survive better in moist and shady environments and tend to grow on the side of trees with the most shade. We know that north of the Tropic of Cancer (in the northern hemisphere), most mosses tend to grow only on the north side of trees because the sun is always to the south, thereby creating a shadow on the north side of trees. Likewise, south of the Tropic of Capricorn (in the southern hemisphere), mosses tend to grow on the shaded south side of trees, since the sun is always to the north. However, in the tropics (in the latitudes between the Tropics of Cancer and Capricorn), the sun may be directly overhead, to the north, or to the south of a tree depending on the time of year, so that no side of the tree is always consistently shaded and the location of the shade varies by season.

I imagine that what side of the tree a moss at the equator grows on would probably depend on where the prevailing shade is coming from and where the most shaded microhabitats of the tree are located.

In other words, at the equator (and elsewhere in the tropics), what matters is likely not where the sun is coming from but rather the specific structure of the tree, the structure of the forest or vegetation surrounding the tree, and what parts of the tree are most shaded as a consequence of their shape, location, and surroundings.

You might expect to find more moss directly beneath branches or epiphytes where they would receive the most shade throughout the day, or on the side of the tree that happens to be in the shade of another nearby tree. In a forest, you would probably expect to see more moss growing lower towards the bases of the trees in the well-shaded under story than near the tops of the trees in the well-lit canopy. Another important factor is local topography. If the tree were sitting on the side of a mountain at the equator, then the entire tree may be shaded at certain times of the year, allowing moss to grow well over the tree at those times.

If a tree at the equator were standing by itself on level ground with no surrounding forest to shade it and were equally lit on all sides, then it may be possible that moss grows on the north side of the tree during the austral summer (when the sun is to the south) and on the south side of the tree during the boreal summer (when the sun is to the north). However, this is just speculation on my part (I'm not a botanist).

I think this seasonal change in the location of moss would require that the moss can either colonize the tree and grow at a faster rate than the switch in seasons occurs, or it can lie dormant in a non-photosynthetic stasis mode to reduce water loss and desiccation stress when it is not in the shade and be able to return to a vegetative state as soon as it is shaded.

That said, there might be some light-tolerant mosses in the tropics that can even grow on well-lit parts of the trees. This seems likely to me because one of the reasons that mosses tend to grow and survive better in the shade is due to reduced water evaporation rates and lowered desiccation stress (i.e., to prevent the mosses from drying out).

In many places in the tropics and along the equator, there is a very high amount of precipitation (although rainfall can vary quite a bit between the wet and dry season in some locations). If precipitation is high enough and consistent enough, then mosses may be able to survive and grow even if occurring in un-shaded areas. Finally, another consideration is the humidity of an area. In the tropics, the air tends to be very humid due to high evaporation and transpiration rates, with very high water vapor content in the air. This high humidity probably also prevents mosses from drying out quickly and may in fact allow them to absorb moisture/water directly from the air. The tropics also tend to be fairly cloudy as a result of the high humidity, especially during the rain season, so cloud cover may frequently shade mosses at the equator and elsewhere in the tropics.