At a cabinet meeting on Tuesday, President Trump said that the Federal Emergency Management Agency had swiftly deployed personnel to Central Texas, as catastrophic floods roared through the region.

“You had people there as fast as anybody’s ever seen,” Mr. Trump told Kristi Noem, who leads the Department of Homeland Security, FEMA’s parent agency.

But FEMA has been slow to deploy certain teams that coordinate response and search-and-rescue efforts, according to half a dozen current and former FEMA officials and disaster experts, most of whom spoke on the condition of anonymity because they were not authorized to comment publicly.

The current and former officials cautioned that every disaster presents unique challenges and that FEMA plays a supporting role to state and local emergency management agencies.

Still, the experts said that the extent of the destruction in Texas, the number of missing people and the complexity of the response would normally trigger a bigger, faster deployment. The death toll in Texas from the floods climbed to at least 111 on Tuesday, with at least 173 people still missing, including 161 in Kerr County, which sustained the worst damage, state officials said.

In an internal update on Tuesday morning, FEMA said it had deployed about 70 search-and-rescue workers to Kerr County and had sent around a dozen others who could help manage responses to Austin. Another unit of about 40 personnel was on standby, able to be in place on short notice.