"With pledges from Texas and Arizona to continue these abhorrent operations indefinitely, the situation is dire, and we consider this a humanitarian crisis - one that could overwhelm our social support network without immediate and sustained federal intervention."
Governor Ducey shot back on Twitter: "Mayor Bowser is lamenting 4,000 migrants - Arizona had 43,570 border encounters in June alone."
Texas Senator Ted Cruz, a Republican, said: "If 4000 is a tipping point, what the hell do you call the three & a half million illegal immigrants who've crossed our southern border?"
Migrants arriving in DC have so far been heavily reliant on local mutual aid organisations for everything from food and assistance to housing and jobs.
Some do not stay long in the nation's capital, reconnecting with family or locating their immigration appointments in other parts of the US.
Washington DC is a sanctuary city, one of many Democratic-led jurisdictions around the country that limits its co-operation with federal immigration law enforcement.
New York City Mayor Eric Adams last week partially blamed the bussing programmes for placing a "real burden" on the city's safety net and shelter system.