  We quantitatively address the following question: for a QFT which is
partially compactified, so as to realize an RG flow from a $D$-dimensional CFT
in the UV to a $d$-dimensional CFT in the IR, how does the entanglement entropy
of a small spherical region probing the UV physics evolve as the size of the
region grows to increasingly probe IR physics? This entails a generalization of
spherical regions to setups without full Lorentz symmetry, and we study the
associated entanglement entropies holographically. We find a tight interplay
between the topology and geometry of the compact space and the evolution of the
entanglement entropy, with universal transitions from `cap' through `bridge'
and `corner' phases, whose features reflect the details of the compact space.
As concrete examples we discuss twisted compactifications of 4d ${\cal N}=4$
SYM on $T^2$, $S^2$ and hyperbolic Riemann surfaces.
