ZHEN’AN, China—For 30 years, Yu Huajian visited villages in rural China to remind couples to have just one child, to abide by the law and help the economy. He also pursued violators of the much-hated policy and oversaw abortions.

Since the one-child policy was abandoned in October, Mr. Yu and some of the half a million other family-planning workers have knocked on rural doors with a different message: How to play with children, read to them and raise them with better skills.

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