What's the proper science-driven way to treat mild scrapes/abrasions (that have a bit of dirt in them) from a mountain biking crash? I tried to research this using search engines and found such a random variety of recommendations. Some were saying to make sure you have your tetanus booster while others were citing a study that shows the antibacterial properties of dirt/clay. It's hard to know who to believe. 

How should one treat common mountain bike crash abrasions on their knee?
Briefly, a quick rinse with clean water, and possibly dressing it, are probably all you need to do on the trail. This assumes a day visit to a typical trail centre or cross country riding near to facilities in a country with a reasonably accessible healthcare system, i.e. most mountain-biking. This was also confirmed by an outdoor first aid course I did recently (these are a very good idea).
You don't know what's in the dirt, so studies discussing antibacterial properties of certain mineral compounds aren't very relevant, even if they're effective against the bacteria that might be an issue, in the wild as opposed to in the lab. As an extreme example, consider coming off into fresh dung on a rock - slippery, unhealthy, but still capable of causing a graze.
Alcohol wipes are now deprecated for wound cleaning; first aid kits now include sterile saline wipes instead, but apart from using a folded corner to flick particles of dirt away, they're not very effective.
Rinsing with your water bottle is about as good as you'll get for cleaning while on the trail. This is a good reason to have plain water in (one of) your bottle(s) - most other drinks would not be such a good idea (similarly, washing muck out of important bike parts). Getting enough flow from a Camelbak isn't so easy, and you need to have plenty of water with you to do this. Some ride leaders I know carry a spare bottle that they don't drink from, for this, dehydrated novices etc., but in general the wounded person's own bottle should be first choice.
Once the wound is fairly clean it may be a good idea to dress it to keep it that way, but on something like a knee that flexes a lot while riding, this can be more trouble than it's worth, though if you're bleeding probably it's a good idea to cover the wound. Large stretch fabric sticking plasters (band aids) are ideal, though in my riding first aid kit I have a selection of these and bigger dressings. I carry more than most people as I'm used to leading kayaking and road riding. A more thorough clean might be needed when you can, e.g. a shower, or several bottles of water when there's a tap.
After a recent tumble leading to unsightly grazes on my knee and elbow (and that first aid course) my preferred dressing for wounds on moving parts is now the sort with a sterile pad and built in conforming bandage - it doesn't rely on adhesion so is more sweat-proof, and flexes well, while looking rather like a cartoon bandage.  Zinc oxide tape is a also good for holding down dressing, if you use plenty.
Keeping your vaccinations up to date (tetanus being the most relevant) is an important general rule, regardless of what you do at the time of the injury.