1) Increasing the average income in both countries

As Charlie comments he has "made both countries richer". Suryasnato Dasgputa answers this:

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.

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, for the reasons above.

2) Decreasing the average income in both countries

Suryasnato continues:

To decrease the average income in both countries, you have to
be a below average earner in Emerald but an above average earner in
Flint. You must live in Flint and then move to Emerald. That way
you will decrease the averages of both countries.

3) 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.

4) 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 one
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 $(n+2)\times 16 000$

Well done Thomas, but you haven't explained where you got these equations from. However with a little algebra, we can see where these came from.

From part $3$ we know that for the average in Emerald to decrease and the average in Flint to increase we need a person with a salary over $£20 000$ to move from Emerald to Flint. To make the average still valid after the move, Emerald must have more than just our citizen who is moving.We then intorduce some symbols for our equations, $F=$average income on Flint, $n=$number of people living on Flint, $I=$the income of the person moving island.

Then the total income on Flint before the move is $n\times F$, so the total after the move is $n\times F+I$. Then after the move the population on Flint is now $n+1$. So the average salary is $\frac{n\times F+I}{n+1}=2\times F$. We can rearrange this to give the salary $I$. Which is $I=2\times F(n+1)-n\times F$, which simplifies to $I=F\times (n+2)$, as given by Thomas above.

So if Flint had a population of $5$ and an average income of $£16 000$, then the migrants income would be $I=16 000\times (5+2)=£112 000$.

5) 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, and not change the other by moving either way, Charlie must have exactly the income amount of either island.

To make one country not have a valid average income, he must be the only citizen of a country. As average is not valid with only one piece of information.

Suryasnato also noted;

Charlie's move would have unexpected results if when he is about to move, a fire hits Flint and all of the below average earners are wiped out. Then, the average income is increased to $£20,000$. So when Charlie moves, he will be a below average earner and make Flint poorer.

True, though we hope a fire wouldn't wipe out half the residents of an island!

Thomas and Surysnato have spotted some interesting changes, can you find any others?