Depends on the job. For instance, I'm moving from an IT job into a ministry job. I accepted a call a couple months ago, and I'm FINALLY moving this weekend. I have been in contact virtually every weekend since, and have traveled there since to set things up. But my situation is perhaps different. I will be assuming leadership of an organization, and will have a fairly high level of visibility.
That may be different from your's depending on the position. If you're one of 10 guys on a team of people in an IT shop with no direct reports and not a lot of leadership decisions to make? There probably isn't a lot of followup needed. But if you are taking a position such as director/ceo, or higher level of leadership, you may want to stay in touch, get updates on how things are going, etc. Even though you're not being paid, you'd be "on the job" and expected to have some sort of transitionary responsibility.