Promoting bile flow is part of the liver cleanse/support/detox regimen, definitely. The main way that the liver sloughs off toxins and waste is through releasing it with the bile, and reduced bile flow means reduced detoxing.
Dandelion and Coptis encourage bile flow.
Milk Thistle does not, it actually sheathes liver cells and helps them metabolize better. It acts on a cellular level and it's very strong, you can save someone from poisoned mushies with MT.
Turmeric has Curcomon, (sp?) which is a powerful anti-oxidant, protects the liver and can actually heal cirrosis, and helps red blood cells become less "sticky" so aids circulation.
Bupleurum helps the liver deal with fat, as does lecithin and carrot. this is very important, especially to those of us who ate the standard american or British diet for awhile. ;)
MSM and B vitamins help the liver break things down such as environmental toxins.
Amino Acids (proteins) help the liver MAKE things such as hormones.