The best way to transport a limping dog is to use a pet carrier or crate. First, ensure the crate is large enough for your dog to move around comfortably, and line it with a blanket or towel for extra cushioning. Additionally, you may want to wrap your dog's injured leg with a bandage before transporting them. Lastly, when carrying the crate, use two hands and support the bottom, so your dog doesn't experience any unnecessary jostling.