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For example, me and 7 of my friends are sitting at a lunch table. 3 other people sit down at the same table. We (the 8 of us) leave that table and and sit down at another. This effectively exiles us due to the 3 people that sat down, but required no force.

We were debating what the word for this would be. I don't believe it is "immigration," because we didn't move to an already existing place.

It seems you've shunned the three you left behind—you have deliberately avoided their company.

Welcome to English Stackexchange, Carson.

Your question is similar, although not identical to this: I need one-word expression for "Ignoring someone intentionally"

The behaviour you are describing sounds like you are ostracising the person or people:

To ostracise, “To exclude (a person) from society or from a community, by not communicating with them or by refusing to acknowledge their presence; to refuse to talk to or associate with.

You could use migrate. The Cambridge Dictionary defines migrate as:

If people migrate, they travel in large numbers to a new place to live temporarily

or this might better suit:

to move from one place to another

There is no perfect term for both the movement and the act of leaving others behind.

For a start, try relocate and relocation.

We relocated to another table.

Our relocation left three people behind.

Merriam-Webster's entry for the verb "relocate":

transitive verb : to locate again : establish or lay out in a new
place

intransitive verb : to move to a new location

Most dictionaries I consulted report "relocation" as the noun form with a corresponding meaning.

One caution: many words and phrases for this kind of movement have negative connotations due to the history of forced population movements or of white flight, the phenomenon of white families moving away from communities to avoid living next to black or immigrant families. Because the noun form does not signal whether people are relocating themselves or being relocated, rhetorical care is warranted. Consider the collocation "forced relocation," which refers to moving "large numbers of people under threat, planned and organized by governmental authorities, armed forces and/or militias." So you'll want to be careful with how you use and modify relocation.