Many places like Carlisle Auto Air sell flush guns ( about $45) to be used with your air compressor ... they also sell recovery kits so you can reuse it.... the pro models run it through multiple times and filter as it goes through the system... you can do that shade tree and do a great job , just takes a little care and time...
I would split my flush into Gross and clean.... use the Gross first on all parts.. then use the clean.... and preferably new clean on each section being flushed the last time....