ROME — At the height of the migration crisis, Italy called on the European Union to keep its promise and help shoulder the burden of the hundreds of thousands of asylum seekers landing on the Continent’s southern Mediterranean border.

Instead, Italy felt the bloc’s cold shoulder, and the ensuing popular frustration helped fuel the rise of the anti-immigrant nationalist Matteo Salvini, who virtually sealed ports to ships carrying rescued migrants before his political overreach led to his unexpected exit as interior minister this summer.

With Mr. Salvini gone, centrist European leaders see a chance to apply the lessons of the past and give fresh momentum to efforts to overhaul the bloc’s immigration system.

On Monday, interior ministers from France, Germany, Italy, Malta and Finland, which holds the rotating presidency of the European Council, struck a temporary deal to redistribute migrants on aid ships to some member states immediately following their rescue.