We are engaged but live far apart or in different countries. How do we keep the relationship going, and what are the limits?

Communication about the marriage is allowed with good manners, but avoid excessive private conversations late at night; involve the families and plan a realistic wedding date.
A fiancé(e) is still not a mahram, and Allah forbids approaching zina by any route (al-Isra' 17:32). Distance does not remove the risk, since video calls and private messages can become "being alone together" online. Suggestions: set the purpose of conversations — preparing for married life, money, where to live, work; let family members know or occasionally join; do not share photos that do not cover the aurat; and avoid an overly long engagement. If one of you will move country after marriage, discuss visas, work and registering the marriage early. See the question on engaged couples talking by phone and Marrying abroad.