if you want to be environmentally sensitive use the basal oil(methylated seed oil) you can certainly use diesel fuel. it is far cheaper and will give you better function at cold weather. You can either apply to a cut stem or apply to the stem of the standing shrub, give it full coverage of 20'' of the lower stem. This method will take alot of chemical and if treating more than a small patch you can easily exceed what the label says you can apply. The LABEL IS THE LAW !!!!
now strategy wise, how much of this stuff are you trying to kill ? that will decide your next move. If you only have a small patch or a few stems, go ahead and basal treat them at your earliest convenience. Strike while the iron is hot !
If you have a big patch or acres of it it is far more efficient to treat it with a foliar application, ( I recommend a 4% mix). If the stems are taller than you can treat, you need to cut them down and then use a foliar spray on the sprout growth mid summer. Buck thorn has the ability to make fruit on sprout growth that develop the same growing season. if your stems are an inch to three in diameter a clearing saw is the way to go. if they are bigger than that a big clearing saw will work but a chainsaw isn't a bad choice either. if they are bigger than 3 inches you are hopefully dealing with just a moderate number per acre. I would go ahead and treat the stumps as you are doing the cutting.
depending on what it costs you to buy glyphosate (active ingredient in roundup) it may be cheaper to use than the garlon. For cut stem use, you must put glyphosate on no more than an hour from the time of cutting. You can this glyphosate down to 50% of 40% plus active ingredient formulations.
the recommended mixes are of the herbicide right out of the jug. Not percent active ingredient.