Moss is normally removed from lawns by scarification  (vigorous rakin)g in the Autumn if they  contain a lot of thatch or in the  Spring  if only a light  scarification is required to prevent the build-up of thatch. The moss removed from the lawn can composted.

However, moss is slow to breakdown having a high lignin content    and it may  be three years or even longer before there are no recognisable pieces of moss left in the bin. It is therefore suggested that it be composted in a separate “moss compost bin” . Askorganic write that moss produces special aromatics and phenols that require the action of specialist bacteria  to break them down and these may not be found in the compost bin but may be present in the soil beneath the  lawn or in the  flower bed. They recommend the adding of soil  to the bins. The microbes require moisture to function  and break down the moss. Others suggest that the moss should be  added to the bin in small amounts e.g. after the spring maintenance scarification. Mixing  the  moss 1:4 with other greens, such as grass mowings, is recommended.

The moss can be stored in a separate bin  and added gradually as other ingredients become available. Moss spores may survive cold  composting but the RHS state that it won’t add significantly to the risk of moss forming in the garden as it will already be widely distributed. Can I compost  It? reports cases of the moss  regrowing  in a  very short time after the compost is spread. To avoid this problem the separate moss compost bin moss should be hot compost bin so that a temperature is reached that will kill the moss. A pallet bin with regular turning for the first four weeks should achieve and maintain sufficient heat particularly if it the aeration is undertaken based on the compost temperature

A Hotbin can be used to compost moss  provided the moss is in relatively small quantities,  is well mixed with other waste and the Hotbin is operating at 40-60 C . It is recommended that dead grass be added at the same time to  compensate for the high lignin content of the moss.  

Sources and further reading

https://www.rhs.org.uk/advice/profile?PID=422

http://www.askorganic.co.uk/composting/Composting%20moss.htm

http://www.can-i-compost-it.com/can-i-compost-moss/

https://www.hotbincomposting.com/blog/scarifying-and-raking-lawns-for-thatch-and-moss.html