Essentially if you get a War Machine from a care package you may use it as long as you don't expend its last round to make it go away. I personally use all the War Machine ammo except the last round. Once you get another care package, use whatever it is and your War Machine will be brought back up. Notice the ammo will be replenished as if you just got another War Machine care package. Enjoy.
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After testing this works on Death Machine as well. Note, you MUST get the War/Death machine from a care package, not a simple death streak.