Car starts, rough idle,engine dies after raising idle speed for 15 or 20 secs

Sounds like a automatic choke sticking....Take the air duct loose from carb..While accelerating engine spray bottle throttle spray in.....as the car tries to die...stop spraying...Repeat 3 or 4 times.....It will clean your injectors. Runs alot better...Remember....bottle throttle spray...not carborator spray

you have a possible vacuum leak somewhere or maybe clogged egr

A 1986 Mustang with a 4 cyl engine should have a carb and a choke. It sounds like the choke is not closing and-or the fast idle cam is not raising the idle speed until the choke opens.

It can be issue with low fuel pressure,the fuel pump is getting old both needs to be checked.Also get the IAC valve checked.
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These details will help.
Thanks.

The first thing to do is not set up the idle speed, but to set the Idle mixture screw to lean best idle setting. First, turn in the mixture screw until the engine dies or runs worse, then back out the screw (recommend turning ¼ to ½ turn at a time). The engine should pick up speed and begin to smooth out.

The battery has nothing to do with the idle speed

Sounds like a possible throttle positioning sensor.

Change speed sensor located near the gearbox.

check timing and set idle to 650rpm

Hi

Try take your car to a local mechanic...It seems your car may need a servicing and a proper check up...There are a series of valves that will need tuning...It is best done by a qualified mechanic.

possible engine coolant temp sender