1) The one saving grace of Yari Ashigaru is the fact that they are cheap. This alleviates financial burden and makes them expendable in just about any situation. If you start out with a very poor clan then Ashigaru can be used to bolster your number of troops. Sometimes, sheer numbers wins battles even if the troops are inferior.
2) Of course if you are Oda in your campaign then Ashigaru actually have a purpose. They come even cheaper then normal and have some honor bonus if built in Owari. So Oda can actually weild Ashigaru with some effectiveness. This is good since his initial lands are vast with lots of borders to defend.
3) They serve as a cheap way to garrison a territory with low loyalty. Instead of spending 200+ to put another unit in a province for a garrison, Ashigaru can be used at half the price.
4) The Bait. Ashigaru can be used to bait armies into switching there positions and even changing there entire plan.
As Takeda I use Ashigaru to open up archer units for attack. What I do is charge an Ashigaru unit at an archer unit. Well usually Archer units have a Yari back up of some sort so they step forward to take care of my charging Ashi unit. Well once they are far enough away from the Archers I flank the Archer unit with my Cavalry. The archers will not run toward there defenders who are attacking my Ashi, instead they will skirmish to a safer place, which just happens to be farther from help for them.
Another way to use Ashigaru as bait is in a bridge assault. When you want to take a bridge you need a way to get the AI to bring their units within your archer range. So you charge an Ashi unit at the bridge and some of the enemy units will counter by coming closer to the bridge. When they do you can start to pummel them with arrows.
5) This one comes from a post on The totalwar.com Community site forum from a guy named Mike Palmer. He calls it "soaking off". Here it is… If I am attacking let's say province B, and there are additional enemy troops in adjacent provinces A and C whom the AI might shift after my move into B to aid the defense, I'll attack provinces A and C with Ashigaru. If the enemy troops stay home, I call off the attack. If the troops do pull out, I walk in and take the place for free. (In my old wargamer days we called this "soaking off.")
6) This tactic comes from Kingsley Gardner email@example.com.
Yari Ashigaru's speed makes them invaluable to chase down routed troops early in the game when you don't have cavalry. They will beat most routed troops as they get a bonus for attacking from behind.
7) They are also good to block bridges when you are defending. Because they are so cheap you don't care about them getting hit from friendly fire.