  We have determined the superconducting and magnetic properties of four
samples of Lu$_x$Zr$_{1-x}$B$_{12}$ ($x=0.04$, $0.07$, $0.17$, and $0.8$) using
muon spin rotation ($\mu$SR) and magnetometry measurements. We observed a
strong magnetic signal in both the $\mu$SR and magnetometry data in one sample
($x=0.07$), likely caused by the formation of static moments of size $\approx
1\,\mu_{\rm B}$ due to a clustering effect of the Lu$^{3+}$ ions. In all other
samples, we find only a small magnetic signal in the $\mu$SR data thought to
originate from boron nuclei in the B$_{12}$ cages. The superconductivity is
found to evolve with $x$, with a decrease in $x$ resulting in an increase in
critical temperature and a decrease of the penetration depth. Most remarkably,
we find the formation of nodes in the superconducting gap for $x \leq 0.17$,
providing a new example of an $s$-to-$d$-wave crossover in a superconductor.
