I travelled over to SA last month and had a couple of internal flights as well as the international ones. I had to collect my rifle at each and every leg then re-book it onto the next one. So even my flight out had to check in at heathrow, collect at joburg, check in again for the flight to East London. Once at East London all my paperwork was checked for the actual hunt for PH registration numbers and outfitter registration etc. Then allowed to leave the airport. Same on my return as I had a couple of days in Durban between finishing hunting and flying from Joburg.
TBH its a right pain in the arse but has to be done. I found heathrow were happy for ammo to be in a locked box, then put in check on luggage. But SA air wanted it out of the hold luggage with a seperate tags.
I also printed off the page from SA air website stating one piece of sporting luggage was allowed INCLUDED in addition to your allowance with no charge. This came in use on the way out of the country as they tried to sting me for £100 for excess baggage. They said they would allow it on this occasion when I said I had their policy printed out in my papers.
So allow extra time for this as you will think you are the first traveller ever to take a firearm overseas. Have a great hunt.