How to share clipboard between Android devices, preferably without need for internet connection
Thanks to Izzy's trove of curated apps
Paste Easy (that can be downloaded from Aptoide) was suggested. Reason being that this app caters to early versions of KitKat which is OP's requirement. I didn't try the app but OP has confirmed that it meets the bill.

For alternatives, see Multi Device Clipboard 
on Izzy's list.