Gravitational wave detectors have a frequency range that they are sensitive to.
In the case of LIGO it is about 10Hz to 1kHz. The lower limit is imposed by seismic noise, the upper limit by "shot noise" (basically not having enough photons to sample the interferometer path difference at high frequencies).
LISA is in space and doesn't have the problem with seismic noise. However, there is still an upper limit determined by shot noise. As I understand it, LISA will not/can not use the same sort of resonant cavity arrangement that LIGO uses to boost the effective laser power and hence number of photons in the apparatus. Thus the upper frequency limit for LISA is more like 1Hz.
So now to your question. What LIGO can see that LISA can not, are gravitational waves with frequencies of 10-1000 Hz. The astronomical phenomena that lead to such waves are the merger of compact binary systems with stellar masses, rapidly rotating, asymmetric pulsars and perhaps supernova explosions.
Lower frequency GWs can only be seen with LISA. This would include stellar binary systems with orbital periods longer than about 10 seconds, merging supermassive black holes and maybe GWs from the big bang.
What could move from being observable in LISA to being observable in LIGO? In principle, the merger of any binary system results in a gradually increasing frequency and amplitude. Stellar mass, neutron star or black hole binaries can get to frequencies around 1kHz before they merge, but sweep through lower frequencies before that, but with much lower amplitudes. Perhaps the best bet would be the merger of intermediate mass black holes with masses of $10^3$ to $10^4$ solar masses, which would have merger frequencies of tens of Hz, but a significant amplitude at much lower frequencies prior to merger.
I suppose the implication of the quote you gave is that this newly discovered inspiralling white dwarf binary might be another possibility. I don't think that is correct. It has a 7 minute orbital period, which puts it in LISAs frequency domain (the GWs emitted have twice the orbital frequency), but because white dwarfs are physically larger than neutron stars, the peak frequency at merger (some time in the future) will only be a few Hz and not visible to LIGO.
