I actually liked Beyond Earth Rising Tide better than CiV.
One of the major aspects that I liked in BE was the use of Diplomatic Capital for agreements and trait upgrades. I would love to see that feature make its way to Civ VI.
Another feature that I think has potential is the affinity system. While I still believe that it could have been fleshed out more fully in BE, I did like the idea of the same civilization going in different directions - these directions shaping not only bonuses but the appearance of the leaders, units, and buildings. In Civ, everything generally starts out with a diverse appearance based upon cultural region, but when we hit the Industrial era, everything starts to look the same with different flags. I would suggest building an affinity system into Civ that starts out early in the game but begins to have major effects around the Industrial era. This could be rooted in the CiV division between Freedom, Order, and Autocracy. Alternatively, they could represent a more broad spread of ideas as in BE: environmental harmony, militarism, and scholasticism . Imagine the different flavors that a single civ might have using these affinities: Harmony Greece might be a nature cult whose priestlike leader wears flowing robes and laurels. Militaristic Greece might be represented by someone like Alexander in full armor. Scholastic Greece might be led by a philosopher like Pericles of Athens. In the late-game, these affinities might significantly affect diplomatic relations and agreements - Harmony civs being environmentalist etc.
The affinity system would also help to represent the differences that we find in the real world between the civilization development on various continents. So-called 'primitive' nations like the Iroquois might appear as they do in the game not because they are backward in technology but because they are Harmony based, while England is Scholastic. And what fun would it be if in another game those affinities were reversed?
I am also a fan of the way that BE allowed for satellites and ocean-based cities. While the implementation could be fine-tuned, these are both areas that were always sorely lacking in earlier Civ games (not including Alpha Centauri). By the time my nation reaches the 21st century, I should have the option of expanding into the water. When I discover satellites, it would be nice if they served a function beyond revealing the entire map.
One final area of BE that I would like to see implemented in Civ is the use of tile-effects. BE's use of miasma was a bit simplistic (it would have been more fun to see different tile-effects for different biomes), but Civ has always seemed oddly neutral about terrain affecting units. I want units in harsh environments to suffer penalties - in tundra, in desert, in jungle. I want to choose civs like the Arabs in part to avoid the desert penalty. I want to choose upgrades to help my troops survive better on tiles that I am looking to expand into.
I think that's it for now - there may be other promising ideas from BE that I just am not thinking of at the moment. For instance, I'm pretty sure that BE is the first title in the series to limit wonders by placing them on their own tiles, which is being continued in Civ VI.