  First we investigate the evolutions of the radius function and its gradient
along the volume-preserving mean curvature flow starting from a tube (of
nonconstant radius) over a compact closed domain of a reflective submanifold in
a symmetric space under certain condition for the radius function. Next, we
prove that the tubeness is preserved along the flow in the case where the
ambient space is a rank one symmetric space of non-compact type (other than a
(real) hyperbolic space) and the reflective submanifold is an invariant
submanifold and the radius function of the initial tube is radial. Furthermore,
in this case, we prove that the flow reaches to the invariant submanifold or it
exists in infinite time and converges to another tube of constant mean
curvature over the invariant submanifold in the $C^{\infty}$-topology in
infinite time.
