Hesitates and shuts off sometimes on acceleration

My mom's car has been doing something weird. Sometimes it will start to hesitate and sometimes shut off on acceleration. This has been going on off and on for five months.
It always idles perfectly and never runs rough or struggles during acceleration. It is only a hesitation right as you first start to push on the accelerator when coming from a stop or from a slow (less than 10 mph) roll. If you're on an incline when it occurs, the engine will shut off. After the hesitation occurs, the car accelerates just fine and you can even do wide open throttle at that point without a hiccup. It also appear to happen after the car has been driven for a bit and is warmed up. It rarely or never happens when the car is cold and not fully warmed up.

I'm not sure if its related but we first started to notice this after I replaced the battery. On the off chance it was some weird battery defect, we replaced the replacement battery but still no change.