  Recently, with the help of Parisi-Sourlas supersymmetry an intriguing
relation was found expressing the four-point scalar conformal block of a
(d-2)-dimensional CFT in terms of a five-term linear combination of blocks of a
d-dimensional CFT, with constant coefficients. We extend this dimensional
reduction relation to all higher-point scalar conformal blocks of arbitrary
topology restricted to scalar exchanges. We show that the constant coefficients
appearing in the finite term higher-point dimensional reduction obey an
interesting factorization property allowing them to be determined in terms of
certain graphical Feynman-like rules and the associated finite set of vertex
and edge factors. Notably, these rules can be fully determined by considering
the explicit power-series representation of just three particular conformal
blocks: the four-point block, the five-point block and the six-point block of
the so-called OPE/snowflake topology. In principle, this method can be applied
to obtain the arbitrary-point dimensional reduction of conformal blocks with
spinning exchanges as well. We also show how to systematically extend the
dimensional reduction relation of conformal partial waves to higher-points.
