The Canadian Mounted Rifles were raised in Canada. The RCD went over to South Africa as Canadian Mounted Rifles and fought successfully to have their name changed to The Royal Canadian Dragoons. Quite a few of The RCD contingent stayed on in South Africa to fight in the newly formed Transvaal police or with Howard's Scouts and subsequently in the Canadian Scouts after Major "Gat" Howard was killed.
"Gat" Howard was an American who was in Canada to sell Gatling Guns, and joined The RCD as a Lieutenant in charge of machine guns to go to South Africa.
The Canadian Mounted Rifles saga is a little complicated. The second contingent which deployed in 1900 included the 1st and 2nd Battalions, Canadian Mounted Rifles. The 1st Battalion was formed around a cadre from the Royal Canadian Dragoons and, as George pointed out, was renamed RCD during its tour. The 2nd Battalion was based on the NWMP and was commanded by the commissioner. Both units were very effective, with the edge going to the RCD. The 2nd CMR had some organizational issues, including having the CO sent home as medically unfit and the original DCO, Major Sam Steele, being transferred to command the Strathconas before the unit embarked for South Africa.
In 1902 the 2nd Regiment Canadian Mounted Rifles saw action in the closing months of the war, while four more regiments, 3rd through 6th, arrived after the war had ended.
Absolutely nothing to do with the current situation, but I thought I best clarify the Boer War issue.