Pulis was unveiled as Albion head coach on Friday to replace the sacked Alan Irvine, and he has wasted no time in making changes at The Hawthorns.
Downing ends a three-year association with the club – that saw him take temporary charge prior to the appointment of Pepe Mel in 2013 – while Kiely's departure comes after eight years at the club.
The 44-year-old made 88 appearances as a player before becoming goalkeeping coach in March 2010, however he will not feature in Pulis' plans.
Kelly's time at the club has been much shorter having arrived as Irvine's assistant in June, and he has followed the Scot out of the door after an inconsistent start to the campaign.
Director of football administration Richard Garlick paid tribute to the trio, and wished them well for the future.
"The club would like to place on record its deep appreciation for the contribution of Keith, Rob and Dean within the coaching structure," he told Albion's official website.
"Keith and Dean have been here a number of years and took over first-team control in a caretaker capacity during a difficult period last season. They played a key role in stabilising the team.
"Rob joined at the start of the season and became a valued member of the coaching set-up.
"Changes in personnel are part of football life and it is obviously sad to say farewell to three popular and respected members of staff.
"We wish to thank them for their efforts and wish them all the very best in their future endeavours."
Who Pulis will bring in as part of his new-look backroom staff remains to be seen, but long-time assistant David Kemp is expected to join after attending Albion games with the Welshman.
