How do I get my dog to follow me in Minecraft while in a minecart?

Have it follow you derp

You first have to get a bone and tame it. Then after you tame it your dog automatically follows you or teleports to you.

As long as you have fed it a bone and it is tamed, it should follow you everywhere. It may look like the dog is falling behind, but if it does fall behind, it should catch up by teleporting back to you.