The Clippers have renounced their rights to Richardson, declining to match the Phoenix Suns' six-year, $42.6 million offer sheet.
Richardson, who reportedly recently became engaged to singer Brandy, signed the contract with the Suns on Thursday evening.
Richardson had been expected to stay with the Clippers, so the move is somewhat of a surprise.
"We're pretty ecstatic," Suns general manager Brian Colangelo told ESPN Insider's Chad Ford. "I think he's going to be a fantastic addition to our team."
The Clippers aren't standing pat, however. They have acquired veteran swingman Kittles and cash from the Nets for a future second-round pick.
The Clippers have been maintaining for a week that they would likely match the offer for Richardson. However, a number of trade scenarios have popped up over the past few days that convinced the Clippers not to match in an effort to preserve their cap space for a deal down the road.
Kittles, who makes $10.3 million this season, is in the last year of his contract. With Kittles and Marko Jaric coming off the books next summer, the Clippers would have max cap room to play with again next year.
Richardson's contract, which stipulates that 70 percent of his first-year salary has to be paid to him on Aug. 15, also proved to be problematic. The contract also includes a 15 percent trade kicker.
The trade takes the Nets out of the salary cap threshold this year, slashing payroll to around $47 million for next season.
Information from The Associated Press was used in this report.