The room-temperature operating semiconductor GaMnN is known to be improved in its magnetic property when a highly conductive precipitate $Mn_3$GaN exists. Therefore, it is useful to investigate the behavior of the precipitate through heat treatments for further improvement of its magnetic property. Furthermore, neutron irradiation may further influence the behavior of the precipitates, and consequently, their effects on the magnetization. With the heat treatment, $Mn_3$GaN decomposed and a new phase of $Mn_3$Ga has generated. The kinetics was accelerated by neutron irradiation, which might generate defects that can help the decomposition of N and/or the formation of $Mn_3$Ga. The increase and decrease of the magnetization of the heat-treated GaMnN thin films were explained consistently by the behavior of the precipitates.