  In a rotationally symmetric space $\oM$ around an axis A (whose precise
definition includes all real space forms), we consider a domain $G$ limited by
two equidistant hypersurfaces orthogonal to A. Let $M \subset \oM$ be a
revolution hypersurface generated by a graph over A, with boundary in $\partial
G$ and orthogonal to it. We study the evolution $M_t$ of $M$ under the
volume-preserving mean curvature flow requiring that the boundary of $M_t$
rests on $\partial G$ and keeps orthogonal to it. We prove that: a) the
generating curve of $M_t$ remains a graph; b) the flow exists while $M_t$ does
not touch the axis of rotation; c) under a suitable hypothesis relating the
enclosed volume and the area of $M$, the flow is defined for every $t\in
[0,\infty[$ and a sequence of hypersurfaces $M_{t_n}$ converges to a revolution
hypersurface of constant mean curvature. Some key points are: i) the results
are true even for ambient spaces with positive curvature, ii) the averaged mean
curvature does not need to be positive and iii) for the proof it is necessary
to carry out a detailed study of the boundary conditions.
