WASHINGTON—Congressional Republicans are crafting proposals to respond to the killing of George Floyd and the ensuing nationwide protests, marking a shift in the party’s stance on regulating police practices and raising the prospect of collaboration with Democrats.

In the sharply divided Congress, the uproar over Mr. Floyd’s death and the underlying issues it exposed has spurred lawmakers to sift through many ideas aimed at curbing abuses by police officers. Mr. Floyd, a 46-year-old black man, was killed after a white police…

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