Many studies of education vouchers have looked at the achievement of children who are given vouchers and who transfer to private schools. Generally these studies have found small but meaningful improvements (e.g. here and here). A voucher program, however, is about much more than transferring students from lousy public schools to better private schools it’s about creating incentives to improve the public schools.
Florida’s Opportunity Scholarship Program rated schools. Students at schools that received an F in multiple years became eligible for a voucher that allowed them to attend a private or higher-rated public school. In Feeling the Florida Heat? (ungated version) a paper sponsored by the Urban Institute Rouse et al. look at what happened at failing schools.