In the early 2370s, their main target was Risa, as they saw it as the main problem. The New Essentialists held a protest rally, calling on Federation citizens' complacency with the paradise they've built, the technology they take for granted and their reliance on Starfleet for protection.
Later, the group committed a mock attack on a group of Risians having dinner, knowing full well that the Risians would not attempt to prosecute. The group successfully sabotaged Risa's weather control system in 2373 and attempted to do the same to the planet's vital seismic regulators.
They were partially assisted for a short while by Lieutenant commander Worf; however, Worf realized their actions were not in the best interest of the Federation, and along with his girlfriend, Jadzia Dax, was able to resolve the situation. (DS9: "Let He Who Is Without Sin...")