China, the engine of the luxury-goods industry, is stalling out as the country’s middle-class consumers rein in spending that has long powered the growth of some of the world’s most upscale labels. The erosion of Chinese demand was cast into relief Tuesday as LVMH Mo?t Hennessy Louis Vuitton, the owner of Louis Vuitton and Dior, said sales in its Asia market—which is dominated by China and excludes Japan—dropped 14% over a three-month period ending on June 30, stripping out the effects of currency fluctuations. The decline was partially offset by spending by Chinese tourists abroad, the company said.