  Light beams with an azimuthal phase dependency of $e^{i\ell\phi}$ have
helical phase fronts and thus carry orbital angular momentum (OAM), a strictly
conserved quantity with propagation. Here, we engineer quasi three-dimensional
(3D) structured light fields and demonstrate unusual scenarios in which OAM can
vary locally in both sign and magnitude along the beam's axis, in a controlled
manner, under free-space propagation. To reveal the underlying mechanisms of
this phenomenon, we perform full modal decomposition and reconstruction of the
generated beams to describe the evolution of their intrinsic OAM and
topological charge with propagation. We show that topological transition and
the associated variation in local OAM rely on the creation, movement, and
annihilation of local vortex charges without disturbing the global net charge
of the beam, thus conserving the global OAM while varying it locally. Our
results may be perceived as an experimental demonstration of the Hilbert Hotel
paradox, while advancing our understanding of topological deformations in
general.
