The Kroll Technocracy is a xenophilic and pacifist conglomerate consisting of 3 different alien mammalian species,They have a high sense of morality and would like to annex earth without bloodshed,How can they achieve their goal and integrate humanity in their stellar federation?
Take the Roman approach.
Until Caesar, the expansion of the Roman territories was largely due to their advanced technology. They would go into "barbarian" lands and build forts which would grow into cities. The roman outposts would have things that the locals had never seen before, including concrete and construction techniques that amazed them. The Romans would then make deals with local Chieftans to send their sons to Rome to be educated. When in Rome, the sons of the chieftans would live in luxury and raised the Roman way, and eventually go back to rule their people with Roman values instilled in them.
Imagine an Alien race doing something similar, but on a grander scale. "You won't have to worry about food, illness, or even have to work as hard as you have. Help us build our cities, and we will give all these gifts to you." Again, send the children to the alien world or a faraway outpost on this one to be educated. Let them grow up thinking that the Aliens are very benevolent, and allow the areas to self govern, with only a small tribute given to the alien home world. After a generation or two, they will have installed local leaders very loyal to them. If there was any rebellion or disobedience, send the local loyal areas to crush rebellious ones, but never send your own troops. Increase the amount of gifts to loyal areas, and take them away from any that showed disloyalty,
With their advanced technology (or even just with technology we real humans have currently) your aliens could rather easily convince the entirety of the human population to believe they were gods/angels/supernatural agents.
They could then convince humans to do anything they liked, including but not limited to:
- Not killing each other.
- Develop science and rationality.
- Learn to love, respect and tolerate each other no matter how different.
After they have learned to do former (aided by yet more advanced technology), the aliens can then explain (or hopefully have humanity understand this on their own) that they are not gods, but people from different worlds.
There is no doubt that the human population will be infinitely greatful (and tolerant) towards the aliens since they did afterall enlighten them. Earth can then be seamlessly annexed. No need for any massive cultural/societal exchange, the people of earth have already been intergrating Kroll values since decades/centuries.
Details are important here. To take over the Earth "without force", and "without bloodshed" are two very different questions!
The aliens are in a technologically superior position, so they are the more powerful side here. The Roman approach, as explained by AngelPray, does indeed rely on force (the known strength of the Roman empire) though it is not obvious.
A war continues until one side can no longer keep up. Thus, the those where the sides have very similar force are the bloodiest. For instance, an improvement of helmets does not necessarily save lives if invented by the weaker side. It only makes the war last longer. To prevent such a long-war situation, either do not use force, or as much force that you possibly can.
Do it Hitler style. He did something that would generally be impossible under normal circumstances- In your world/book you could create some sort of conflict that requires immediate attention.
Hitler became a dictator not by means of military, but by means of his people, Hitler became a dictator because his people wanted to give him the power. The great depression was an immense problem for the people of the 1900s, and Hitler told them that he could solve all of their problems. He was smart, not stupid, the aliens should be able to take his example and conquer the medieval lands. Rule with an iron fist cloaked in a velvet glove, or something like that- Ceasar.
You'll want to make the aliens seem like they have good intentions, and they very well may have good intentions, you never disclosed that information. People in the middle ages weren't exactly as 'informed' as we are today, and they would likely be willing to believe pretty much anything.