I haven't gone over every civilization and the possible leaders, but I have a few suggestions.....
Each civilization will have 3 different leaders for the different eras. One of the earliest of the leaders of the civilization will lead in the Ancient and Classical era, a leader from the civilization's middle period will lead in the Medieval and Renaissance eras, and one of the latest (or last) leaders from that civilization will lead in the Industrial, Modern, and, if added, Future eras. I am basing Civ VI after its predecessor, Civ V. Here's some of my ideas...
America:
Ancient and Classical eras- George Washington
Medieval and Renaissance eras- Abraham Lincoln
Industrial, Modern, and Future eras- Franklin D. Roosevelt
Rome:
Ancient and Classical eras- Julius Caesar
Medieval and Renaissance eras- Augustus Caesar
Industrial, Modern, and Future eras- Constantine (the Western Roman Empire didn't fall during Constantine's reign, and before he split the empire, he ruled both east and west, which means he didn't rule the Byzantine Empire, even after the split, because it wasn't called the Byzantine Empire yet, but the Eastern Roman Empire)
England:
Ancient and Classical eras- Arthur (although he is a mythological figure, some historians believe an actual King Arthur existed in the 5th and 6th centuries A.D.)
Medieval and Renaissance eras- Elizabeth I
Modern, Industrial, and Future eras- Winston Churchill
Russia:
Ancient and Classical eras- Ivan IV
Medieval and Renaissance eras- Catherine the Great
Industrial, Modern, and Future eras- Vladimir Lenin
(The Russian unique unit will no longer be the Cossack, but instead the Red Army Infantry, replacing the Infantry. It will have a combat bonus in friendly lands and a combat bonus when other Red Army Infantry units are in adjacent tiles.)
China:
Ancient and Classical eras- Qin Shi Huang
Medieval and Renaissance eras- Zhu Yuangzhang
Industrial, Modern, and Future eras- Sun Yat-sen
Japan:
Ancient and Classical eras- Jimmu
Medieval and Renaissance eras- Tokugawa Ieyasu
Industrial, Modern, and Future eras- Meiji
France:
Ancient and Classical eras- Charles Martel (although he was the de facto ruler of Francia, the lands he ruled were to be France, and therefore, he ruled France)
Medieval and Renaissance eras- Napoleon Bonaparte
Industrial, Modern, and Future eras- Charles de Gaulle
I will think up more later....