A laser-engraveable layer is used in a laser-engraveable element to provide relief images. The resulting laser-engraved elements can take various forms and can be used to apply various inks to receiver materials in an imagewise fashion. The laser-engraveable element has a thermally crosslinked laser-engraveable layer containing a crosslinked elastomeric fluoropolymer and a fluoro-functionalized near-infrared radiation absorber. The crosslinked fluoropolymer has a glass transition temperature of less than and including 0° C. and the crosslinked laser-engraveable layer is formed from a reactive fluoropolymer composition comprising a reactive fluoropolymer, the fluoro-functionalized near-infrared radiation absorber, and a compound that causes crosslinking of the reactive fluoropolymer during thermal curing. The reactive fluoropolymer can have a reactive group selected from the group consisting of: poly(tetrafluoroethylene oxide-co-difluoromethylene oxide) α,ω-diol bis(2,3-dihydroxypropyl ether), poly(tetrafluoroethylene oxide-co-difluoromethylene oxide) α,ω-diol, ethoxylated poly(tetrafluoroethylene oxide-co-difluoromethylene oxide) α,ω-diol, and a multifuncational (meth)acrylate end-functionalized derivative of one of such compounds.