How to serve a motion/petition from across the country?

You are going to have to do some research. Read the statutes governing expungement in that state. You can see them at you local law library. I would be shocked if it requires personal service, but you will have to read the statutes to be sure. If it doesn't say, then ordinary mail is probably OK, but certified will prevent questions of when something was received. If it does require person service, then get in the phone book for that town and find a process server. Hire that process server.