= Solution
Let $G_n(z_1,\ldots,z_n)$ denote the ordered vacuum correlator. Inserting the vacuum-preserving <Möbius transformation> operator and its inverse gives
$$
G_n(z_1,\ldots,z_n)=\prod_i\gamma'(z_i)^{h_i}\,G_n(\gamma(z_1),\ldots,\gamma(z_n)).
$$
For $\gamma(z)=z+\epsilon v(z)$, the three vector fields $v=1,z,z^2$ yield the <global conformal Ward identities for chiral correlators>
$$
\sum_i\partial_iG_n=0,\qquad
\sum_i(z_i\partial_i+h_i)G_n=0,\qquad
\sum_i(z_i^2\partial_i+2h_i z_i)G_n=0.
$$
All formulas below concern separated insertion points and a consistent local choice of branches for fractional powers.
For one insertion, the <translation> identity makes $G_1$ constant and the <dilation> identity gives $h_1G_1=0$. Thus
$$
\boxed{G_1=C_1\text{ if }h_1=0;\quad G_1=0\text{ if }h_1\ne0}.
$$
For two insertions, <translation> invariance gives $G_2=f(z_{12})$, where $z_{ij}=z_i-z_j$. <Dilations> imply $z_{12}f'+(h_1+h_2)f=0$, so $f=C_{12}z_{12}^{-h_1-h_2}$. Substituting into the special conformal identity leaves $(h_1-h_2)z_{12}f=0$. Therefore
$$
\boxed{G_2=\frac{C_{12}}{z_{12}^{2h}},\quad h_1=h_2=h},
$$
with $G_2=0$ when the weights differ.
For three insertions write $G_3=C_{123}z_{12}^{-a}z_{13}^{-b}z_{23}^{-c}$. Under a <Möbius transformation>, $\gamma(z_i)-\gamma(z_j)=z_{ij}\sqrt{\gamma'(z_i)\gamma'(z_j)}$ with compatible local branches. The covariance condition therefore fixes
$$
a+b=2h_1,\qquad a+c=2h_2,\qquad b+c=2h_3.
$$
Solving gives
$$
\boxed{G_3=\frac{C_{123}}{z_{12}^{h_1+h_2-h_3}z_{13}^{h_1+h_3-h_2}z_{23}^{h_2+h_3-h_1}}}.
$$
This is the complete solution because <Möbius transformations> act transitively on ordered triples of distinct points: the ratio of any two covariant nonzero solutions has no remaining invariant variable and is constant. \b[Three-point covariance imposes no equality or triangle inequality among the weights]. Additional fusion rules or other symmetries may force $C_{123}=0$, but none follow from the stated covariance and vacuum assumptions alone.
For more than three generic points, three positions may be fixed by a <Möbius transformation>, leaving $n-3$ independent complex <conformal cross-ratios>. Correlators consequently have a covariant prefactor multiplied by an arbitrary function of these invariants. For example one convenient four-point form is
$$
G_4=\frac{(z_{24}/z_{14})^{h_1-h_2}(z_{14}/z_{13})^{h_3-h_4}}{z_{12}^{h_1+h_2}z_{34}^{h_3+h_4}}F(\chi),\qquad
\chi=\frac{z_{12}z_{34}}{z_{13}z_{24}}.
$$
The <Möbius invariance of the cross-ratio> guarantees covariance for any $F$. Thus \b[global conformal symmetry fixes one-, two- and three-point functions up to constants, but does not fix higher-point functions]. Their remaining functions encode dynamics.
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