All chargers works, just depends on cable and not chargers or kitkat. You get faster(slwoer charging depends on charger and quality of the cable.
For u question, I used Trent 11000 mah external battery pack, 5V 1Amp and with 3rd part usb cable, that are only for charging, without "data pins". It charging at 500 mah when screen is on, screen off at 930-980 mah and working as normal.
I dont know where u get that kitkat have anything to do with 3rd part chargers doesnt works.
Well my third party chargers stop charging when the screen turns off. Even if I use the USB cable that came with the Note. However, old chargers of HTC or Motorola phones do work with any USB cable. And I have read this from several other users as well.
I'm on stock 4.4.2 with stock recovery. What recovery/kernel/ROM do you use?