Ballistol and water mix, "moose milk." I'd add a bit more Ballistol than 20%. I believe Ballistol is mineral oil, but don't know.
I flush my MLs with warm soapy water, let it set for about 5 minutes, then apply moose milk. After drying out the moose milk with cleaning patches, I spray WD 40 to displace water. This works good on ML guns, don't see any reason why it wouldn't work on a break action gun.
Ballistol is good stuff. They make a BP cleaning substance; can't remember the name of it, but it's supposed to be really effective.