  We study non-Abelian vortex strings in ${\mathcal N}=2$ supersymmetric QCD
with the gauge group U$(N)$ deformed by the mass $\mu$ of the adjoint matter.
This deformation breaks ${\mathcal N}=2$ supersymmetry down to ${\mathcal N}=1$
and in the limit of large $\mu$ the theory flows to ${\mathcal N}=1$ QCD.
Non-Abelian strings in addition to translational zero modes have orientation
moduli. In the ${\mathcal N}=2$ limit of small $\mu$ the dynamics of
orientational moduli is described by the two dimensional $CP(N-1)$ model for
QCD with $N_f=N$ flavors of quark hypermultiplets. For the case of $N_f>N$ the
non-Abelian string becomes semilocal developing additional size moduli which
modify the effective two dimensional $\sigma$-model on the string making its
target space non-compact. In this paper we consider the $\mu$-deformed theory
with $N_f>N$ eventually making $\mu$ large. We show that size moduli develop a
potential that forces the string transverse size to shrink. Eventually in the
large $\mu$ limit size moduli decouple and the effective theory on the string
reduces to the $CP(N-1)$ model. We also comment on physics of confined
monopoles.
