  ${Z}_2$-Yukawa-QCD models are a minimalistic model class with a Yukawa and a
QCD-like gauge sector that exhibits a regime with asymptotic freedom in all its
marginal couplings in standard perturbation theory. We discover the existence
of further asymptotically free trajectories for these models by exploiting
generalized boundary conditions. We construct such trajectories as quasi-fixed
points for the Higgs potential within different approximation schemes. We
substantiate our findings first in an effective-field-theory approach, and
obtain a comprehensive picture using the functional renormalization group. We
infer the existence of scaling solutions also by means of a
weak-Yukawa-coupling expansion in the ultraviolet. In the same regime, we
discuss the stability of the quasi-fixed point solutions for large field
amplitudes. We provide further evidence for such asymptotically free theories
by numerical studies using pseudo-spectral and shooting methods.
