I had a friend who was practicing with his new .416 for his upcoming Cape Buffalo hunt. He was using it to shoot out of his second story window at ground squirrels in his pasture. Any squirrels that he missed got new holes to hide in.
To specifically answer your question, it is my understanding that the .22-250 was originally developed as a varmint cartridge, and with the right bullets it has been successfully used on deer sized (and larger) animals. I've shot both mule deer and antelope with mine. I have friends who have shot elk and buffalo with theirs.
And the .25-05 was originally developed as a deer cartridge that would also work well on long range shots on varmints on windy days. I've used my .257 AI, which is balistically similar to the .25-06, on everything from prairie dogs to elk. It's my favorite deer, antelope, and bighorn sheep rifle.
So you have been thinking about making the trek out west for a DIY big game hunt. The pronghorn antelope is an excellent choice for your first western big game hunt. It is a good choice because it offers the first-timer an introduction to DIY western hunting with a high probability of success and without the exertion or need for detailed planning that a mule deer or elk hunt might require. While there are several states that you could conduct your pronghorn antelope hunt, I...