Murty has been told the Gers hotseat is his until the end of the season.
But he has now watched his side lose four of the ten games he has presided since being put in charge after Pedro Caixinha’s sacking.
Declan John appeared to have given the former youth team coach the perfect start when he stroked Rangers ahead at Rugby Park six minutes before the break.
But the Light Blues were barraged after the break and eventually succumbed to Kris Boyd’s late double.
“The problem is the basic stuff,” admitted Murty. “When we do that well we look a good team. When we’re sloppy, we invite pressure.
“We invite other teams to exploit areas and that’s what it was today. Killie were better than us at the basic stuff.
“We didn’t pass the ball well enough. We didn’t deal with balls loaded into the box with enough authority at times. I’m looking at the squad and expecting a level of understanding and competence. At times we show it and they’re on the right level - but we manage somehow to shoot ourselves in the foot.”
Murty has been told he will be in charge of the club’s January transfer dealings, with up to four new faces expected to arrive.
But the new boss warned his side that those already at the club are fighting for their futures - and even himself.
• READ MORE: Kilmarnock 2 - 1 Rangers: How the players rated
Murty - who decided not to play defender Bruno Alves after the Portuguese international reported an injury in the build-up to the match - said: “The recruitment work is ongoing and targets have been identified.
“But before that happens behind the scenes we need to make sure the current players understand their roles and execute them.
“When you’re filling a shirt for Rangers you have to perform regardless of the pressure you’re feeling. It has to be worthy of this club.
“You’re always playing for your future as a footballer, chasing your next contract. But you have to earn the right to play because you’re being judged.
“I’ve no doubt that I’m being judged right now by many people.
“And the players likewise. If you want to be at this football club you must have broad shoulders to cope with that.”
Killie have now won four of their last five games and manager Steve Clarke believes his side are on a roll.
“All I’ve said to the players is just about maximising the moment when things are going for you,” he said. “And at the moment things are going for us. You have to try and make that moment last as long as possible.
“It’s a bit like Celtic with their 69 games - I’m not saying we’re going to do 69 games but it’s all about maximising that moment. When the ball seems to be bouncing for you, you have to get as many points as you can.”