The Liberal Democrats, who have backed a committee system as a way of making key decisions rather than the one-party cabinet, have warned against “pulling up the drawbridge”.
Greens have cautioned against not listening to other groups and driving ahead with controversial schemes.
And the Holme Valley Independents said the election result – with Labour winning three seats to give it 36 in the council chamber – was a return to 2018 and that some years would pass before there might be significant change.
One of the seats gained by Labour was in Golcar, where Lib Dem veteran Robert Iredale lost to returning Labour man Graham Turner by just 49 votes. That gives the Lib Dems just eight members.
Group leader Coun John Lawson (Cleckheaton) urged Kirklees Labour to challenge what he called “this catastrophic Conservative government”, to not be “lazy” and not to repeat “Tory platitudes” when meeting high-level ministers such as Michael Gove.
He said the council needed to tackle the cost of living crisis, poor air quality, children’s mental health, road and pavement repairs “and many other things that are simply drifting”.
He added: “The way decisions are made here needs to change.
"We have championed a new committee system approach that gives more communities a voice and makes decisions more robust. The Liberal Democrats will continue to strive for that change.
“Labour have the democratic mandate to form the administration, that isn’t in question, but do we really want business as usual or are the people who voted Labour as well as everyone else in the borough entitled to something better?
“My plea to the Labour leadership would be to make use of your mandate to improve services and tackle what must be dealt with, use it to reach out to other parties in good faith and, please, don’t relapse in the old ways of pulling up the drawbridge.”
The Greens’ Coun Andrew Cooper (Newsome) outlined his concerns that projects that might have been “on hold” pending the results of the election may now go ahead, such as the controversial widening of the A629 Halifax Road in Edgerton, where 126 mature trees are to be chopped down.
He also highlighted greenfield sites in council ownership that could be under threat if they are brought forward for development.
He said: “There are also the missed opportunities. A minority Labour administration might have been more willing to take on board ideas from outside their party.
"On tackling climate change Kirklees has not been a leader but has focused on pilot projects rather than mainstreaming policies that embed good practice.
"However I’m ever hopeful and will seek to engage positively with the Labour administration.”
Charles Greaves, lead member of the Holme Valley Independents, saw his three-man group shrink to two after Terry Lyons lost his seat to the Conservatives.
He backed the Lib Dems’ calls for a committee system, which he described as “the only alternative to one political party dominating Kirklees” and that scrapping the cabinet system would allow all groups to set policy, such as at Bristol and Wirral councils.
He said: “[Creating a modern committee system] would finally give everyone a voice and real chance to make their case.
"The Holme Valley Independents will continue to focus on the issues and get on with the jobs. We have lots of schemes in the pipeline and we are keen to see them delivered.
“If the Government finally comes good on its promises to level up the north, and to give Kirklees a fair funding settlement, we should have the money to do the work and to finally ease the burden on ratepayers.
"It would be great to finally have a year when council tax bills don’t go up.”
Coun David Hall (Liversedge and Gomersal), leader of the Conservative group, said he was “obviously very disappointed” that Labour had achieved a majority and would take leadership of the council again.
He lamented the loss of Donna Bellamy in Colne Valley but welcomed Tony McGrath in Holme Valley North.
He added: “Our Conservative councillors will continue with their excellent work holding this unlistening administration to account.
"We will stick to the values we have always held, and fight at every opportunity for a fairer, more efficient council with better services across the board.”