Is Charlie right?
John considered an extreme case:
Only one person lives on Emerald Isle, other that Charlie, and they earn £22,000
In this case, it is clear that the average income in Emerald increases when Charlie leaves. In general, even if there are many more people in Emerald, we can think of the average income as the amount of money each person would get if all of the money was shared out equally. So with less than the average income, Charlie is leaving with less than his 'fair share', and there is more to go around for everyone else.

Suryasnato Dasgputa said:
Charlie has increased the average income of both countries. In Emerald, he was a below average earner so he was decreasing the average. So once he moved, the average for Emerald increased. When he was in Flint, Charlie was an above average earner so he increased the average in Flint when he moved in.

Increasing the average income in both countries
Suryasnato continued:
In general to increase the average in both countries, someone with an income between £16,000 and £20,000 (but not exactly either) needs to move from Emerald to Flint.

Decreasing the average income in both countries
Suryasnato also explained how the average income in both countries could be decreased:
To decrease the average income in both countries, you must live in Flint and move to Emerald, and be an above average earner in Flint but a below average earner in Emerald (with an income between £16,000 and £20,000). 

Increasing one whilst decreasing the other
Jeremy from Longston Middle School wrote:
There are four ways of this happening.
1) Charlie earns over £20,000, and moves from Emerald to Flint. Then Emerald's average goes down and Flint's goes up.
2) Charlie earns over £20,000 and moves from Flint to Emerald. Then Flint's average goes down and Emerald's goes up.
3) Charlie earns less than £16,000 and moves from Emerald to Flint. Then Emerald's average goes up and Flint's goes down.
4) Charlie earns less than £16,000 and moves from Flint to Emerald. Then Flint's average goes up and Emerald's average goes down.

Moving from Emerald to Flint, reducing the average in Emerald and doubling the average in Flint
Thomas Hu from A Y Jackson school answered this:
The only way he could decrease the average income of Emerald and double that of Flint is to move from Emerald to Flint with an average income of $(2+n)\times a$, with $n =$ number of people in Flint, and $a$ being the average income in Flint. Now let the incomes of Emerald and Flint back to their original values, £20,000 and £16,000. Thus Charlie must have an income of $(n+2)\times 16 000$

It is easy to see how this works by imagining average income as the amount of money that each person would get if the money were shared out equally. To double the average income in Flint, each person's share must be doubled, so increased by £16,000. This would require $ £16000\times n$, where there are $n$ people living in Flint.

The population of Flint will also increase by one when the new wealthy resident arrives. So the total amount of money earned in Flint must be enough for this new person to also get their fair share of $ £2\times16000= £32000$.

So the new person must earn $ £(16000n+32000)= £(n+2)\times16000$, as Thomas found.

If someone who earned $ £(16000n+32000)$ lived in Emerald, then they can't have been the only resident of Emerald, or the average income in Emerald would have been $ £(16000n+32000),$ which is at least £32,000 and so more than £20,000. So at least one other person lives in Emerald. If all the money earned in Emerald were shared out equally, each person in Emerald would get $ £20,000$, so this wealthy person who earns $( £16000n+32000)$ contributes far more than their fair share. So the average income in Emerald will fall when they leave.

Other interesting circumstances
Thomas Hu from A Y Jackson school noticed some other interesting changes in average income that could occur:
To change the average income of one country, and not change the average income of the other, when moving either way, Charlie's income must be the same as the average income of one of the countries.