The first thing to know about Nitrogen both in living plants and the soil is that it is Nitrate or nitrite that we are concerned with.
Free nitrogen (the gas, which is what we are talking about when we say nitrogen) is pulled in by bacterium and converted to nitrates, this is then further processed into nitrites which is the usual form plants can use.
Dormant or dead grasses will contain 3/4 less nitrite than green, living grasses, this is, as has been mentioned, because the plant moved as many nutrients as it could to the
roots in preparation for dormancy.
Green, living grasses that are cut and allowed to dry will contain about 1/2 the nitrate and nitrite of the freshly cut, still green and moisturized grass.
For composting you can use grass in any stage, the fresh cut will provide the best conditions for heating up a compost heap since it contains the most nitrite and the most moisture.
Green cut and then dried grass will provide nitrite but will not help in the heating of a heap.
Dead cut (cut dormant or dead grass) will act more like a "Brown" component in a heap.