CUMMINS ISX TURBO COUGH AND THEN BLACK SMOKE PUFF

are you getting fault codes? Black smoke means over fueling,,could be a bad airflow sensor ,,go to a shop,,with a cummins you do not have to go a dealer,there are many shops that can hook up a computer and find the problem and many small shops are cummins authorized do better work and are much cheaper than dealers.
Only go to a dealer when it is under warranty or you have no other choice.
If you run Utah there is a great shop that is also a Cummins service dealer in Aurora Utah next to Salina UT,,best cummins shop I know of,,they know there stuff,,they are in the phone book,, Mason Diesel Service

 

We had a truck that did exactly what you describe, If I remember right what was happening was the splines on the turbo actuator were worn enough to where the drive pressure could move the vanes enough to where it would lose boost. had to replace the turbo and actuator.

EDIT:
Just re-read your first post, being a 07 you probably have a CM870, which uses a bit different style turbo and actuator than a CM871 which is what I was talking about.

Only thing that comes right to mind on a CM870 would be to make sure the air line going from the drivers side of the engine over to the turbo actuator doesn't have and holes rubbed in it or anything, the actuator control boxes like to leak sometimes too (front of the engine on the drivers side, just above the oil pan, actuator air line runs into it)

 

My guess it's EGR valve. Disconnect it and see if it changed anything.

 

The coughing symptom is usually intake air backing up and reversing direction and coming back out the intake. This is usually due to exhaust gases getting backed up and not flowing through the turbo. Like the others mentioned, have the turbo actuator and system checked.

 

I am also having a Cough noise followed by a puff of black smoke in my 2006 International with and ISX. its every once in a while. Maybe a couple times in a day. Most times in a pull, loaded but not always. Normally no fault code. But today I have two actually.
It only has about 60,000 miles on a complete overhaul. However, I have never been impressed with is performance
since I got it back. Had more power before I had it done actually. I have unplugged the EGR to see if it made any difference and it does not seem to. Any advice what to check or have checked? Thanks in advance!