How could the Kingdom of Lotharingia continue to exist throughout history in Europe?
While it is true that the West and East Franks fought over Middle Francia - which is not the same as Lothariniga - this did not begin until Lothair II and Louis II died without any legitimate heirs. It's important to keep in mind that Lothair was the eldest son of Louis the Pious, so in effect he received the best Frankish lands - the rich areas along the Rhine and Rhone, as well as Italy. But Lothair quickly left Italy to his son Louis (II), ruling mostly in Lotharingia (which included Burgundy). This proved defendable, and Lothair was able to maintain the integrity of his lands even as they were divided between his three sons following their father's death: Louis II receiving Italy and the Imperial title, Lothair II becoming King of Lotharingia, and Charles becoming King of Provence. Charles was the first to die, and his realm was divided up between Lothair and Louis.A better definition of Lotharingia would help.
It was geographically such a nightmare it wouldn't possibly last. It was simply a middle-strip between Germany and France, the Lowlands, and Northern Italy, divided by the Alps. Not necessarily an easy group of territories to govern in the middle ages. If it was simply limited to the Lowlands and possibly Lorraine is might be a bit more feasible..