The 10,000-mile rally from the capital to Ulan Bator is scheduled to take four weeks, with the ambulance then being donated in Mongolia to serve the local community.
The graduates - Tak Lo, Ashok Ananthnarayan, Nalin Gupta and Giacomo Tortorapnatedsi - began the trip on 23 July.
The rally has been organised to raise funds for two nominated charities, the John Smith Memorial Trust and the Christina Noble Children's Foundation.
Tak Lo said: "It comes down to what I learned at business school...it gives one the necessary confidence to tackle life's issues. The confidence to stick to one's values. And that has to be the best lesson, because for me, the value of a tested life is immense."
The students have drawn on many skills learned during their MBA in organising the expedition, including dealing with multiple stakeholders and managing the planning process.