  The next Galactic core-collapse supernova (CCSN) will be a unique opportunity
to study within a fully multi-messenger approach the explosion mechanism
responsible for the formation of neutron stars and stellar-mass black holes.
State-of-the-art numerical simulations of those events reveal the complexity of
the gravitational-wave emission which is highly stochastic. This challenges the
possibility to infer the properties of the compact remnant and of its
progenitor using the information encoded in the waveforms. In this paper we
take further steps in a program we recently initiated to overcome those
difficulties. In particular we show how oscillation modes of the proto-neutron
star, highly visible in the gravitational-wave signal, can be used to
reconstruct the time evolution of their physical properties. Extending our
previous work where only the information from a single detector was used we
here describe a new data-analysis pipeline that coherently combines
gravitational-wave detectors' data and infers the time evolution of a
combination of the mass and radius of the compact remnant. The performance of
the method is estimated employing waveforms from 2D and 3D CCSN simulations
covering a progenitor mass range between 11$\mathrm{M_{\odot}}$\, and
40$\mathrm{M_{\odot}}$\, and different equations of state for both a network of
up to five second-generation detectors and the proposed third-generation
detectors Einstein Telescope and Cosmic Explorer. Our study shows that it will
be possible to infer PNS properties for CCSN events occurring in the vicinity
of the Milky Way, up to the Large Magellanic Cloud, with the current generation
of gravitational-wave detectors.
