Short answers, pulled from the story.
Gian Galeazzo Visconti received the title of Duke of Milan in 1395. This event transformed a city-state into a formal duchy and marked a turning point for the family that had been shaping Northern Italy for decades before that moment.
Francesco Sforza conquered Milan in 1450 after marrying Bianca Maria Visconti, daughter of Filippo Maria. This union ended the brief Ambrosian Republic and established House of Sforza rule that would last until 1535.
Ludovico il Moro died in exile at the Castle of Loches on the 27th of May 1508. He was later captured at Novara in 1500 after being driven out by Louis XII who claimed Milan through Valentina Visconti.
Philip II received an imperial diploma signed in Brussels on the 11th of October 1540 granting him control over Milan. His rule began officially when Henry II of France recognized his claim through the Treaty of Cateau-Cambrésis in 1559.
Napoleon Bonaparte conquered northern Italy in 1796 bringing an end to centuries of ducal rule. An interim government junta managed affairs before the Treaty of Campo Formio ceded Milan to France in 1797.