Wont start when key in ing. and turning on than cutting off if battery dead

Factory alarm, when manually unlocking the door the lights flash & horn usually go off & the car will start & shut off within a second or 2. Have 2 96 jeeps that do it the panic button on the keyless entry is the only way to shut it off. If the battery has died & a new one is installed it trips the panic mode as well. If the battery is dead but vehicle starts with a jump but shuts off soon after jumper cables are removed, the alternator is bad & cant keep the computer powered up & will shut down the vehicle. A little more specific with symptoms will make diagnosing much easier.