Falzone, a former communications instructor at the University of Pennsylvania, started it in 2011.
This time it will be Moshe Kahlon, a former communications minister and Likud member.
Tim Miller, a former communications director for Jeb Bush's presidential campaign, called Mr. Trump's comments "inhuman".
Her replacement will be Risa B. Heller, a former communications director for Senator Charles E. Schumer.
Stephen Conroy, a former communications minister, warned that a protracted contest risked making Labor into a "laughingstock".
Donald A. Baer, a former communications director for Mr. Clinton, will take over as chief executive, the firm said.
Paolo Gentiloni, a leading opposition member and a former communications minister, said Internet regulation was inevitably political.
Priti Patel, employment minister Patel is a former communications chief for the Referendum party and a staunch Eurosceptic.
"I subscribe to that," said Cheri Jimenez, 31, a former communications director for the conservative organization Concerned Women for America.
"It brings him joy," said Risa Heller, a former communications director, one of more than a dozen former aides who recounted his sayings, often while imitating his voice.
Jim Dyke, a former communications director for the Republican National Committee unaffiliated with any candidate, said: "This is the nail in the coffin in her campaign.
