You were right to be concerned! It can be tough to try and "think" like your pump in order to troubleshoot.
We use Alaris pumps, and they, like stated above, are gravity based.
On our pumps, you can set up you primary fluid and tubing with a rate of KVO or whatever, and then connect your secondary to the highest port on the primary tubing, and hang it higher.
Then, you can go into the pump's main screen, choose secondary, put in your volume and rate. Then all the clamps are opened, and you can see a small picture on the screen that shows the smaller, higher-up bag "highlighted" (it's darker in the little picture than the primary, big bag).
I've made it habit to always stand and wait and make sure the piggyback bag is dripping.
Also, these pumps will run the word "secondary" along the screen, and make a very loud Beep when the secondary infusion is done and the pump is reverting over to the primary fluid/rate.
I haven't had any issues so far!
Hope that helps!