I just moved entire website to a new server, but am still getting traffic to my old one (caching issue?)

I just sold a website, so I transferred everything to new owners' server. The name servers are now pointing to his server as well. I also replaced all of my Adsense tags with the buyers'. However, I'm still getting a small number of daily Adsense views to all of my tags even though none of them are on the site anymore (1,000 page views per day or so).

My question is this: is there a caching issue here or something? Are DNS providers not honoring the new server location and will it just take time for this issue to go away?

How can I resolve this issue?