TBILISI, Georgia—Aleko Grigolashvili was just 11 years old when he and a friend ran away from home to join Georgian fighters battling Moscow-backed rebels in Abkhazia, a separatist enclave on the Black Sea.The adventure barely lasted a month in the early 1990s, before Aleko returned to his panicking grandparents, and to school. But it set the tone for a lifetime of war against Russia. In 2008, Mr. Grigolashvili, by then an officer in the U.S.-trained Georgian military, fought Kremlin-backed separatists and regular Russian troops in South Ossetia, another breakaway region of Georgia. He was wounded in the knee.So when Russia invaded Ukraine last year, using a script honed in Georgia, Mr. Grigolashvili went to war again. “He saw it all as one war, our war,” recalled his wife, Christine Svanidze. On a reconnaissance foot patrol in December, near a dreary east Ukrainian town named Happiness, Mr. Grigolashvili was ambushed and killed.