Managing inter-district school choice, by Yuchiro Kamada and Fuhito Kojima

 Here's a paper that considers inter-district school choice, motivated by Tokyo day care centers.  I think  a similar problem arises in the EU in allocating foreign study opportunities for college students.

 Ekkyo Matching: How To Connect Separate Matching Markets For Welfare Improvement  By Yuichiro Kamada And Fuhito Kojima

Abstract. "We consider a school-choice matching model that allows for inter-district transfer of students, with the “balancedness” constraint: each student and school belongs to a region, and a matching is said to be balanced if, for each region, the outflow of students from that region to other regions is equal to the inflow of students from the latter to the former. Using a directed bipartite graph defined on students and schools, we characterize the set of Pareto efficient matchings among those that are individually rational, balanced and fair. We also provide a polynomial-time algorithm to compute such matchings. The outcome of this algorithm weakly improves student welfare upon the one induced when each region independently organizes a standard matching mechanism"

" In Japan ... allocation of slots at accredited daycares are conducted by individual municipal governments and, with few exceptions, a child can only attend a daycare in the municipality of their residence. The City of Tokyo, for example, is divided into 23 small municipalities ... and each conducts a matching independently. Due to the small sizes of the regions, many families would find inter-district admissions—which is called the ekkyo admission ... to be a viable option. Moreover, as a large metropolitan area, many people cross a city boundary to commute, making it potentially more convenient to put their children to a daycare center close to their workplace"