  Next-generation gravitational-wave detectors will provide unprecedented
sensitivity to inspiraling binary neutron stars and black holes, enabling
detections at the peak of star formation and beyond. However, the signals from
these systems will last much longer than those in current detectors, and
overlap in both time and frequency, leading to increased computational cost to
search for them with standard matched filtering analyses, and a higher
probability that they are observed in the presence of non-Gaussian noise. We
therefore present a method to search for gravitational waves from compact
binary inspirals in next-generation detectors that is computationally efficient
and robust against gaps in data collection and noise non-stationarities. Our
method finds tracks in the time/frequency plane of the detector that uniquely
describe specific inspiraling systems. We find that we could detect $\sim 5$
overlapping, intermediate-strength signals (matched-filter signal-to-noise
ratio $\rho\approx 58$) without a sensitivity loss. Additionally, we
demonstrate that our method can enable multi-messenger astronomy: using only
low frequencies ($2-20$ Hz), we could warn astronomers $\sim 2.5$ hours before
a GW170817-like merger at 40 Mpc and provide a sky localization of $\sim 20$
deg$^2$ using only one ``L'' of Einstein Telescope. Additionally, assuming that
primordial black holes exist, we derive projected constraints on the fraction
of dark matter they could compose, $f_{\rm PBH}\sim 10^{-6}-10^{-4}$, for $\sim
1-0.1M_\odot$ equal-mass systems, respectively, using a rate suppression factor
$f_{\rm sup}=2.5\times 10^{-3}$. Comparing matched filtering searches to our
proposed method at a fixed sensitivity, we find a factor of $\sim10-50$
speed-up when we begin an analysis at a frequency of 5 Hz up to 12 Hz for a
system with a chirp mass between $\sim[1,2]M_\odot$.
