The external battery packs are acting like the plug in charger.
Many phones chargers just put out 5 vdc which runs the internal charger regulator in phone. Some phone charger are current limited 4.4vdc to 5 vdc supply which the internal phone charger regulator just does a pulsed switch connection. This reduces heating from charger series pass MOSFET within phone.
The smart phones, with their larger capacity batteries usually use the current limited external charger supply. Original iPhone used a 1200 mA-hr LiIon. Newer models are likely in 1500 mA-hr range. The iPhone original charger was just a USB cord plugged into a 5 vdc, 1 amp current limited switching power supply. This made it somewhat compatible with charging from computer USB port, although max current spec on USB is 500 mA's. Desktop USB will likely give you 1 amp, laptops are more likely to limit it to 500 mA. You will be good putting a low drop out 5 vdc @ 1 amp regulator fed from 4 series connected AA alkalines ( or larger cells).
Buying an external battery pack to feed a phone charger input is pot luck. It might have nothing more then a resistor in series with the external battery which much be higher then 4.5 vdc to be effective. A good quality one has a current limited output switcher that can milk the last drop of blood out of four AA alkaline batteries. For an iPhone, 4 AA's total capacity will give you about one full recharge.
You don't want to put a fully charged battery in parallel with a discharged battery. You can get excessive charging current that can damage battery. If phone relies on a current limited external charger you can damage phone charging circuit putting 4 series connected AA batteries directly to external charging contacts.