Jelly, 16 bytes
Ḣ;J€×J+ŻƲ{\×"⁸SƲ
A monadic Link that accepts a list of integers, the monomial coefficients in ascending order and yields the falling factorial coefficients in ascending order.
Try it online! Or see the test-suite (showing I/O in descending order).
Note: a constant polynomial (an input of length one) yields an additional, redundant zero coefficient at \$x_1\$, I'm guessing that's acceptable, if not Ḣ;ȧJ€×J+ŻƲ{\×"⁸SƲ$ would work for \$18\$ bytes.
How?
Effectively employs the recurrence relation of Stirling numbers of the second kind, \$S(n, k)\$:
$$S(n+1, k) = k S(n, k) + S(n, k-1) \space \vert \space 0 \lt k \lt n$$ $$S(n, 0) = S(0, k) = 0$$ $$S(n, n) = 1$$
Ḣ;J€×J+ŻƲ{\×"⁸SƲ - Link: list of integers, C
Ḣ - behead C -> removes the constant term, N, from C
Ʋ - last four links as a monad - f(Rest)
J€ - range of length of each -> [[1], [1], ..., [1]]
\ - cumulative reduce by:
Ʋ{ - last four links as a monad - f(Current (initially [1])):
J - range of length -> [1..length(Current)]
× - {Current} multiplied by {that} (vectorises) -> M
Ż - prefix {Current} with a zero
+ - {M} add {[0]+Current} (vectorises) -> next Current
×"⁸ - {that} zipwise multiplied by {Rest}
S - column-wise sum
; - {N} concatenate {that}