  Using results of the thermodynamic Bethe Ansatz approach and conformal
perturbation theory we argue that the $\phi_{1,3}$-perturbation of a unitary
minimal $(1+1)$-dimensional conformal field theory (CFT) in the $D$-series of
modular invariant partition functions induces a renormalization group (RG) flow
to the next-lower model in the $D$-series. An exception is the first model in
the series, the 3-state Potts CFT, which under the $\ZZ_2$-even
$\phi_{1,3}$-perturbation flows to the tricritical Ising CFT, the second model
in the $A$-series. We present arguments that in the $A$-series flow
corresponding to this exceptional case, interpolating between the tetracritical
and the tricritical Ising CFT, the IR fixed point is approached from ``exactly
the opposite direction''. Our results indicate how (most of) the relevant
conformal fields evolve from the UV to the IR CFT.
