Republican lawmakers in at least six states are pushing to eliminate the death penalty, signaling a broader reversal by many conservatives on an issue that has long been a bedrock for their party.

In Kansas, Kentucky, Missouri, Montana and Wyoming, Republicans in legislatures controlled by their party have sponsored bills this year to end capital punishment, citing fiscal and moral concerns. Some Republicans in New Hampshire, where Democrats dominate the statehouse, are backing a similar proposal. And a GOP lawmaker in Louisiana’s...