In the Kansas system, judges are appointed by the governor, but they face what has traditionally been a pro forma retention vote on the ballot after their first year, and every six years thereafter. Across more than half a century, until the past few years, the retention system was nonpolitical, with no judges rejected. But this year is decidedly different. This time Republican officials expect that a flood of out-of-state political money — untraceable under loose campaign laws — will be drawn to the fight.

Four of the five judges who face retention elections in November have been part of court majorities that struck down aspects of Mr. Brownback’s disastrous tax-cutting program, which caused huge budget cuts to public education, state roads and other basic services. The fifth judge on the ballot is a Brownback appointee. Two other judges, targeted by conservative groups in 2014, narrowly survived retention votes.

“We’ve seen an unprecedented assault on the judiciary for the past number of years,” said former Gov. Kathleen Sebelius, a Democrat. She and the other former governors, including Mike Hayden, a Republican, and John Carlin, a Democrat, began a campaign this week to stop the court purge.

Judges are particularly vulnerable to the kind of attacks and distortions common in big-money political campaigns because of their independence — the very quality that is key to maintaining the rule of law.

Upon taking office in 2011, Mr. Brownback promised to succeed with “a red-state model” of the trickle-down myth that big upper-bracket tax cuts somehow promote economic growth. Instead, revenues plummeted, resulting in an education funding crisis that threatened schools with closings. Voters reacted to this sham last month in primary elections in which 14 Brownback loyalists were ousted from the Legislature by insurgent moderate Republicans, the rarest of political creatures.