I think the issue involves the new low battery mode algorithm or power protection circuit and older or defective batteries. Similar to an old/dying car battery, as it fails it loses cranking amps. You can open the car door and interior lights and tones will work, but if you try to put a load on the battery, (use the starter,) the amperage requirement is too high and the battery will not turn over the starter.
Power-Cycling
In the case of the phone, heaviest draw occurs on transmit not iOS loading... So when an application tries to access internet, or in the case of a incoming or outgoing call, or a cell site ping, the TX draw causes a dip in voltage, or amperage. The Active voltage/amperage protection sees the dip and forces a power-cycle. (Like a power dip on any computer power supply.)
Sluggishness
Since the introduction of low battery mode, iPhone will lower power draining processes, such as high backlight, and non-critical applications from TX/RX in the background. I also believe the processor can throttle or underclock to save battery. This effectively slows the processor and will cause applications to slow significantly.
Signs of battery defect
Unusually fast charging/discharging
Battery percentages that change irregularly
Battery going from 30% to < 5% in short time while using
Powering off @ less than 5%
Powering off @ any percentage
Symptoms of defective battery
Won't charge
Won't Power on
Excessive heat on discharge
Erratic battery percentages
Screen Dimming
Hangs
Sluggishness
Reboots
Loops
Broken LCD (Because your frustration will eventually lead to throwing the phone hard against walls/floors/apple display tables/etc.)
My example
With my iPhone 6+, if I try different aftermarket batteries, some just don't charge or will discharge quickly. Some defective ones will charge ok, but will cause weird symptoms. (Like the car battery example above.) Backlight will dim quickly, (a symptom of backlight low power mode.) applications will slow down, (a symptom of processor low power mode,) and the device will powercycle. (a symptom of low battery/improper voltage/amperage detection.)
This also make a lot of sense as to why we're reading about the iPhone 6s battery replacement campaign...