WARSAW, Poland – The 3-year-old was too sick to leave his hotel room.

He has had congenital heart disease since he was born. A second surgery was already planned for when he turned three.

Then the war started.

Now, as the young Ukrainian lay in his hotel bed 500 miles from home, his oxygen levels dipped dangerously low. His pulse raced. He'd spent the night feverish and throwing up. His lips and fingertips had turned purple.

Downstairs, a volunteer doctor hustled from patient to patient in the hotel storage room being used as a makeshift doctor's office. As Dr. Mary Tao listened to children's...