The Boers used a wide variety of firearms throughout their various wars, with both the Native Africans and the British. The Boer Republics had no firearm manufacturing capabilities, and got them wherever they could. During the time of the Zulu Wars and First Boer War (1879 and 1880-1881), the preferred weapon (in no particular order) was the 500-450 Wesley-Richards falling-block (AKA "Free State Martini"), the W-R "Monkey Tail" capping breechloader, 577-450 Martini-Henry, and .577 Snider. A small number of 11mm 1871 Mausers. There were some (few) documented use of Rolling Blocks, Trapdoors and Winchesters from the USA, as well. Many muzzle loaders were still in use, but the convenience and rapidity-of-fire of metallic cartridges was realized early on. Handguns were probably mostly of European origin, most likely Belgian and British. Colt sold firearms in England, and many made their way to the Cape, Natal, and Transvaal. Adams revolvers, both C&B and cartridge, Webley and Enfield revolvers could quite possibly be "liberated" from British prisoners or casualties.
The Second War, 1899-1902 saw extensive use of the Mauser Model 1893 and 1895, in 7MM, and 1889/1891 Mausers in 7.65mm. Some still had their Martinis from the previous War, but usually used it to acquire a Long Lee from the British. Handguns from this conflict would probably run the same as before: Webley, Enfield, a few Colts, and Belgian knock-offs. French Ordnance, and Nagants would not be out of place. The Boers used whatever weapons they could buy/smuggle/capture/trade.