It has been my standard practice for decades to clean with warm water, followed by a WD40-saturated patch and then two dry patches, followed by a heavy application of RIG. I run one, or perhaps two, dry patches down the bore prior to the next shooting session and have never had a either corrosion problem or a fouling problem. I use spit patches on the range and, even though I do not clean between shots, fouling is so minimal that it is a non-issue. I can't say I buy the admonishment to not mix BP and petroleum products.
Another vote for RIG