Leak on the wheel.
My son is on his 2002 GS traveling/ moving from Geneva to Munich and was calling me while on a fuel stop. The trip WAS going well and he asked me what to look at on the bike and I said check your tires and see if anything is leaking. He said\ cursed there was a bit of oil on his tire. Less than a year ago he called me while on a trip telling me there was a horrible noise coming from the rear end and oil. Some how we\he got it back home and several dollars later the BMW dealer in Geneva repaired the final drive. He had the bike in for service recently and asked them to change out the oil in the final drive thinking it would be good to get it flushed. While talking to him on the phone I asked him to check for free play and noise which he said that checked ok. He got his tool kit out and took out the fill plug and he says the oil shows on the threads. If the final drive was over filled will they seep out of the seal. I would think it would be coming out of the vent. He was unable to purchase gear oil at the fuel stop so I said get the thickest engine oil they got and go on, he didn't add any. He drove for a short time and rechecked and said hardly and oil the bottom of the final drive so he is moving on. What are his odds of making it another 250 miles?
He just called and said he is entering Austria and seems to be ok. I asked him about the oil level when he first checked at the first fuel stop and he said when he took out the fill plug it puked out some oil out of the fill plug.
Get it checked out by a pro asap and get the right oil (and amount) in there.
The problem may as simple as a leaky crush washer from the last service.