During the "Age of Discovery" (say, prior to WW I), there were long expeditions (often a few years) to explore hitherto-uncharted parts of the globe, but the only one I can think of where one party overtook another was the "race" of Scott and Amundsen to the South Pole. Of course, both parties knew that the other was doing the same thing, starting at close to the same time.
I can only imagine the state of mind of Scott when he finally reached the geographic South Pole and saw a Norwegian flag already there, with a polite letter from Amundsen.
There were three important personalities involved in Antarctic expeditions at that time:
Scott (Royal Navy) was incompetent: he thought he could learn skiing when he got to Antarctica. He saw no reason to adopt barbaric practices from uncivilized indigenous peoples who lived in cold regions of the globe. There was an important lecture at the Royal Geographic Society about how to navigate at extreme latitudes, attended by both Scott and Amundsen, but only Amundsen learned anything.
Amundsen (Norwegian private citizen) was so competent as to be boring. He had previous experience with arctic conditions in Norway and Greenland, had been on one Antarctic expedition already, and found useful materials and practices from the locals. He did not lose any of his company.
Shackleton (Merchant Marine, not involved directly in the South Pole part of the expedition) is the interesting character: he learned from his mistakes and also did not lose a man, even in his later failed expedition.