Thank you for your question about moving moles. Unfortunately, moving moles is about as challenging as moving homeless camps. They may be here to stay. There are traps (which you don't want) and poisons (that you don't want), and then there are a few (and largely ineffective) deterrents.
There are few/no scientific methods recommended to rid your property of these (rather interesting, and often helpful) vertebrates. But, know that your neighbor(s) may not appreciate your technique. Here's a link to some suggestions.