Fed up with underwriting the nation’s broken federal immigration system, Miami-Dade County plans to stop paying the cost of temporarily housing undocumented immigrants in its jails. The dramatic shift in policy comes at a time when the cash-strapped county is coping with a tight budget, but some county commissioners say they are also calling attention to what they say is a serious human-rights issue. “Not only is it about saving money,” said County Commissioner Sally Heyman, a Democrat in a nonpartisan post. “It’s about saving people.” At issue are Immigration and Customs Enforcement’s requests to keep prisoners in custody for 48 hours after they are scheduled to be released so that they can be turned over to federal authorities. Detainees are often deported soon after.