Dyer was the recipient of a letter containing racist abuse, which is being investigated by police, in the wake of Saturday's defeat by Livingston, which was Kilmarnock's eighth in nine Scottish Premiership matches.
The Kilmarnock manager freshened up his attacking personnel by bringing in Danny Whitehall, Mitch Pinnock and Greg Kiltie and the three of them combined for Kiltie to net the opener four minutes before half-time.
Whitehall then netted his first Premiership goal from the spot in the 53rd minute after Mark O'Hara had been sent off for bringing him down.
There was no way back for Motherwell, who are now without a win in nine matches stretching back to Halloween.
Motherwell made a bright start but had to wait 20 minutes before they started creating a series of decent chances.
Devante Cole threatened from an overhead kick and a header, Allan Campbell fired inches wide of the far post from the edge of the box and Ricki Lamie headed on to the roof of the net.
Rory McKenzie had a shot from distance easily held at the other end while Motherwell's first effort on target came just before the opener. Campbell got a good contact on Liam Grimshaw's near-post cross but Danny Rogers palmed away the midfielder's header.
Killie were ahead within 60 seconds. Whitehall played the ball inside from the left flank and Pinnock laid it off to Kiltie, who guided the ball inside the post from just outside the box.
Carson hurt his knee when landing after clutching a cross but initially played on after receiving treatment.
The Northern Ireland international soon found himself facing a spot-kick after O'Hara hauled Whitehall to the ground as the striker turned him to latch on to Pinnock's lay-off.
Referee Greg Aitken showed the red card and Whitehall delivered an emphatic finish from 12 yards.
Carson soon had to scramble to save Whitehall's free-kick but the goalkeeper was clearly struggling and was soon replaced by Aaron Chapman. Motherwell left-back Jake Carroll also came on for his first appearance since rupturing an Achilles tendon 10 months ago.
Killie lost Calum Waters to injury but they could have added to their lead.
Chapman saved from Aaron Tshibola after the ball fell kindly for the unmarked midfielder 12 yards out and Kirk Broadfoot headed against the bar after the substitute goalkeeper had failed to collect a corner.