  We present a theoretical study of coherent quantum dynamics of a single
magnetic fluxon (MF) trapped in Josephson junction parallel arrays (JJPAs) with
large kinetic inductances. The MF is the topological excitation carrying one
quantum of magnetic flux, $\Phi_0$. The MF is quantitatively described as the
$2\pi$-kink in the distribution of Josephson phases, and for JJPAs with high
kinetic inductances the characteristic length of such distribution ("the size"
of MF) is drastically reduced. Characterizing such MFs by the Josephson phases
of three consecutive Josephson junctions we analyse the various coherent
macroscopic quantum effects in the MF quantum dynamics. In particular, we
obtain the MF energy band originating from the coherent quantum tunnelling of a
single MF between adjacent cells of JJPAs. The dependencies of the band width
$\Delta$ on the Josephson coupling energy $E_J$, charging energy $E_C$ and the
inductive energy of a cell $E_L$, are studied in detail. In long linear JJPAs
the coherent quantum dynamics of MF demonstrates decaying quantum oscillations
with characteristic frequency $f_{qb}=\Delta/h$. In short annular JJPAs the
coherent quantum dynamics of MF displays complex oscillations controlled by the
Aharonov-Casher phase $\chi \propto V_g$, where $V_g$ is an externally applied
gate voltage. In the presence of externally applied dc bias, $I$, a weakly
incoherent dynamics of quantum MF is realized in the form of macroscopic Bloch
oscillations leading to a typical "nose" current-voltage characteristics of
JJPAs. As ac current with frequency $f$ is applied the current-voltage
characteristics displays a set of equidistant current steps at $I_n=2en f$.
