If I visit a friends island to invite his villager to live with me, will my friend lose that villager?

Solution 1:

In New Horizons, the only way to swap villagers between islands works as follows.

Let's use Julian as an example, who lives on the island of Alfa. In order for Julian to move to Beta, the following must happen, in order.

Day 1

1. The Alfa player encounters Julian with thought bubbles above his head. When talked to, he admits that he's thinking about moving out. Alfa's player can encourage him to stay, or to follow his dreams and move out.

Day 2

2. If convinced to move out, Julian will be in "packing" mode the next day. (All furniture converted to cardboard boxes). This is colloquially known in the community as being "in boxes".
3. The Beta player visits Alfa via Online or Local play, talks to Julian, and suggests Julian move to Beta. (This requires an empty plot)

Day 3

4. Julian is no longer present on Alfa, and a Julian is moving in on Beta.

Conclusion

It is possible to get friends' villagers to move to your island, but only if your friend encourages them to leave first! But at the same time, it doesn't seem possible to "duplicate" villagers in this way -- they follow a strict "one-out, one-in" policy.