It’s not entirely clear, but from context, the Lyseni Spring seems to have involved a popular uprising against the Rogare family in Lys and an elite coup against the Rogare family in Westeros, partially because of the overweening power of the Rogare family which threatened the oligarchical republic and partially because the Iron Bank does not like competition:

“In the end, it was Larra Rogare and her wealthy, ambitious family who helped break the power of the regents and, almost certainly, that of Lord Peake. It was an inadvertent role they played, however, caught up as they were in the Lyseni Spring. This was a time when the Rogare Bank waxed greater than the Iron Bank, and so fell prey to the plots to control the king; they were blamed for many more acts than they were actually guilty of. Lord Rowan, then the Hand and one of the last regents, was accused of being complicit in their crimes and was tortured for information. Ser Marston Waters, now somehow Hand of the King in his place (Munkun, the only regent at this time besides Rowan, is reticent to discuss this in the True Telling), dispatched men to seize Lady Larra after having arrested her brothers. But the king and his brother refused to give her up, and were besieged in Maegor’s Holdfast by Waters and his supporters for eighteen days. The conspiracy eventually unraveled as Ser Marston—perhaps recalling his duty—attempted to fulfill his king’s command to arrest those who had falsely implicated the Rogares and Lord Rowan. Waters himself was killed by his own sworn brother, Ser Mervyn Flowers, when he attempted to arrest him.

… the Lady Larra Rogare of Lys. She was a great beauty of Valyrian descent, and seven years the prince’s elder when she wed him at nine-and-ten. Her father, Lysandro Rogare, was the head of a wealthy banking family whose power waxed even greater following the alliance to the Targaryens. Lysandro assumed the style of First Magister for Life, and men spoke of him as Lysandro the Magnificent. But he and his brother Drazenko, the Prince Consort of Dorne, died within a day of one another, beginning the precipitous fall of the Rogares both in Lys and the Seven Kingdoms.

Lysandro’s heir, Lysaro, spent vast sums in pursuit of power and fell afoul of the other magisters, even as his siblings became embroiled in plots to control the Iron Throne. After his fall, Lysaro Rogare was scourged to death at the Temple of Trade by those he had wronged. His siblings received less fatal punishments, and one among them—Moredo Rogare, the soldier who carried the Valyrian sword Truth—eventually led an army against Lys.” (WOIAF)

So this reads a lot like the rebellion against Piero II Medici after the death of Lorenzo Il Magnifico, which led to the expulsion of the Medici from Florence, combined with a bit of the Pazzi Conspiracy, combined by some of the intrigues during the regency of Henry VI. However, there’s a lot of detail we’re missing here about how the Rogare tried to control the Iron Throne, what role the Peakes played in all this, what Marston Waters was about, and so on. 

Hopefully when Fire and Blood Volume I comes out, we’ll get a fuller explanation of this incredibly complicated binational political event.