How could my world's transplanted human population reestablish contact with Earth in as stable of a manner as possible

Question

Over the course of millennia, some of the people of Earth have been transported to entirely new worlds. Over time, they reestablish civilizations, expand, grow culturally and scientifically, and eventually discover ways of reaching the other planets their fellow humans have washed upon.

However, their ability to contact other planets raises one important point: Might it be possible to 'go back home'? Historical artifacts and memory generally prove that these people did not originate from the world they came from, so could they rediscover their 'homeland' and return to it.

My current answer is yes, it is physically possible for transport and communication between these outside worlds and Earth. However, I haven't figured out just how these other humans would go about doing it. Those on Earth aren't aware of their existence, bar a small number who these outsiders could use to establish communications.

In all, my question is: How would a nation/political entity as advanced as our own go carry out the process of initiating communication with the nations and people of Earth.

Some things about to note:

• They are only able to send a small delegation to Earth public
• There are some groups on Earth who are aware of them, and who wish to keep them secret from the rest of Earth, due to their ability to use magic
• They are able to communicate verbally and through written words with Earth humans

Edit: I am not asking about the science of the matter, but about the politics, or the actions that a government would take when initiating contact.

Answer 1

A good reference point here is a show called Stargate: SG-1 (I'm bad at links so tell me if this didn't work), which has a lot of first-contact scenarios.

First, we need to look at what Earth has that this nation doesn't; maybe it's knowledge about their past, maybe it's food, maybe it's natural resource, or maybe it's tech.

We also need to see if the public already knows about the fact that they have ancestors on another planet, if they do, there may be mass panic, or there may be a massive public decision to find out about their past.

Alright, so let's imagine a scenario:

The Commonwealth of Technotopia

Technotopia is slightly more technologically advanced than Earth, but they don't have many resources. They'll want to trade technology for Earth resources and due to this, they'll slowly reveal information about Earth to the general public. They'll do this to slowly get the population to accept the possibility that they came from somewhere else, in order to not cause chaos and riots.

The Kingdom of Resourcia

Resourcia is rich in natural resources but poor in technology, much like modern-day Kazakhstan. Opposite to Techtopia, they would offer resources in return for technology. They might not have the option to slowly reveal info, because due to their lack of technology sending out info is difficult. There might be riots, but these are already a common occurrence so it's not too difficult.

The United Provinces

The United Provinces: a US clone. They would probably start by hiding the knowledge in order to profit and quickly rise to dominance over other nations. Eventually knowledge would leak out, and there might be riots and panic, but at that point the UP would already have dominance over other countries.