I am looking for a place that has many different flavors of frozen yogurt, but that mixes in those flavors with either plain or chocolate yogurt (those are usually the only 2 base flavors). Usually this is done via this big machine that smooshes together the base yogurt and whatever flavor you pick
Here is Boston there are a couple of Middle Eastern restaurants that do this (Kiraz, Angora).
Is there something similar in DC and surrounding area?
I know some seasonal ice cream stands have "Flavorbust" frozen yogurt which seem to be what you are looking for. I haven't seen it specifically here, but will keep my eyes open.
Check out Cone E. Island on Pennsylvania Avenue in Foggy Bottom. They have new, soft flavors all the time that can be mixed. One of my favorites is peanut butter and vanilla swirl. They also have a variety of hard ice cream flavors.
Also check out Dickey's Frozen Custard in Farragut West. They have soft yogurt that can be mixed with chocolate and vanilla as well. Probably not as many flavors as Cone E. Island, but good nonetheless.
Hope this helps!