In Schottenloher's mathematically-oriented CFT textbook, "A Mathematical Introduction to Conformal Field Theory," Proposition 9.8 on page 160 states the conformal Ward identities for 2D CFTs as follows: For all $m \in \mathbb{Z}$, for any primary fields $\phi_j$ with scaling dimensions $h_j$,
$$ 0 = \sum_{j=1}^n (z_j^{m+1} \partial_{z_j} + (m+1) h_j z_j^m ) \langle \phi_1(z_1) \ldots \phi_j(z_n) \rangle.$$
For the special case of $m \in \{-1,0,1\}$, these are the Ward identities corresponding to global conformal transformations; see e.g. Di Francesco ("yellow book"), Eq. 5.51.
However, the identity is claimed for all $m \in \mathbb{Z}$, and for $m \not \in \{-1,0,1\}$ the formula would appear to be wrong. For instance, when applied to a two-point function $\langle \phi(z_1) \phi(z_2)\rangle = \frac{1}{(z_1-z_2)^{2h}}$, the formula holds only for $m \in \{-1,0,1\}$.
(It's not a typo when Schottenloher claims the formula for all $m \in \mathbb{Z}$; cf. his Proposition 9.5.)
Is Schottenloher simply wrong about this major point? The statements in this section are proven from formal CFT axioms, but unfortunately the proof is largely omitted for this particular claim (Proposition 9.8). The entire book is careful to distinguish the consequences of the global conformal group versus local conformal algebra (see e.g. discussion at beginning of Section 9.3), which is why I would be surprised by the error.
Best Answer
The equation you quote cannot hold for any integer $m$. A function of finitely many variables cannot obey infinitely many independent PDEs!
There are infinitely many local Ward identities but they involve Virasoro descendant fields. See for example Eq. (2.2.15) in my review article: https://arxiv.org/abs/1406.4290