  The bulk-boundary correspondence relates quantized edge states to bulk
topological invariants in topological phases of matter. In one-dimensional
symmetry-protected topological systems (SPTs), quantized topological Thouless
pumps directly reveal this principle and provide a sound mathematical
foundation. Symmetry-protected higher-order topological phases of matter
(HOSPTs) also feature a bulk-boundary correspondence, but its connection to
quantized charge transport remains elusive. Here we show that quantized
Thouless pumps connecting $C_4$-symmetric HOSPTs can be described by a tuple of
four Chern numbers that measure quantized bulk charge transport in a
direction-dependent fashion. Moreover, this tuple of Chern numbers allows to
predict the sign and value of fractional corner charges in the HOSPTs. We show
that the topologically non-trivial phase can be characterized by both
quadrupole and dipole configurations, shedding new light on current debates
about the multi-pole nature of the HOSPT bulk. By employing corner-periodic
boundary conditions, we generalize Restas's theory to HOSPTs. Our approach
provides a simple framework for understanding topological invariants of general
HOSPTs and paves the way for an in-depth description of future dynamical
experiments.
