We show that a partially hyperbolic $C^1$-diffeomorphism $f : M \to M$ with a uniformly compact $f$-invariant center foliation $F^c$ is dynamically coherent. Further, the induced homeomorphism $F : M/F^c \to M/F^c$ on the quotient space of the center foliation has the shadowing property, i. e. for every $\varepsilon> 0$ there exists $\delta > 0$ such that every $\delta$--pseudo-orbit of center leaves is $\varepsilon$-shadowed by an orbit of center leaves. Although the shadowing orbit is not necessarily unique, we prove the density of periodic center leaves inside the chain recurrent set of the quotient dynamics. Other interesting properties of the quotient dynamics are also discussed.