The Besh”t is to have proven the power of hashgacha pratis to his talmidim. He had them follow an individual leaf which feel from a tree and watched as it landed atop a worm trapped out in a sunny field to provide it a little shade. I could not find the exact makor for this, but you can find it referenced here, here, here, here (this one takes patience), here (#2) – etc. etc.
Contrast with the Sefer HaChinuch in Parshas Metzora (169), which I quote -
"There are groups of people who think hashem's hashgacha encompases everything in the world, all living creatures [I.e. animals] and all other things, meaning nothing moves in this world without G-d decreeing so, so that they think if a leaf falls from a tree it is because G-d decreed for it to fall, and it is impossible for it to fall a second earlier or later than that appointed time. This concept is far removed from intelligence (rachok harbeh min haseichel)."
The Chinuch goes on to say that to deny hashgacha completely is also wrong. The correct philosophy is that there is a general providence, "hashgacha klalit", on all living things, so that no species should become extinct, but not on particular creatures. (Incidentally, even before the advent of the theory of evolution, this is why dinosaur bones cause such a ruckus even to Christian theologians, who shared a similar view re: extinction – here was evidence of mass extinctions having occurred contrary to this view of the Chinuch and other rishonim.) The Rambam in Moreh Nevuchim takes essentially the same view: hasgacha klalit applies to all mankind, but he excludes animals completely. I discussed this with a talmid chacham over Shabbos who could not offhand name a single rishon who held of the broad view of hashgacha the Besh"t spoke of.
Can anyone explain how and why chassidus just dropped the approach of the rishonim on this issue (and please don't just give me the mareh makom to the Rebbe's sicha - I do not fully understand it, so you have to explain it if you do)?