To show that the process always ends after a finite number of moves and to determine all possible configurations where people can end up standing, we will proceed as follows:

1. Invariant of the Average Position:
- Initially, each person stands on a whole number from $0$ to $2022$. The average position of all people is:
$$\text{Average} = \frac{0 + 1 + 2 + \cdots + 2022}{2023} = \frac{\frac{2022 \cdot 2023}{2}}{2023} = 1011$$
- Each move involves two people moving towards each other by 1 unit each. This does not change the sum of their positions, and hence the average position remains $1011$.

2. **Sum of Distances $S$:**
- Let $S$ be the sum of distances between every pair of people. We will show that $S$ decreases with each move.
- Consider two people $A$ and $B$ standing at positions $a$ and $b$ respectively, with $a + 2 \leq b$. They move to $a+1$ and $b-1$ respectively.

3. **Effect on $S$:**
- We categorize the effect on $S$ based on the positions of other people relative to $A$ and $B$:
- **People standing on $a$ or less:** Their distance to $A$ increases by 1, but their distance to $B$ decreases by 1. Thus, their contribution to $S$ is unchanged.
- **People standing on $b$ or greater:** Their distance to $A$ decreases by 1, but their distance to $B$ increases by 1. Thus, their contribution to $S$ is unchanged.
- **People standing in $(a, b)$:** Their distance to $A$ decreases by 1, and their distance to $B$ decreases by 1. Thus, their contribution to $S$ decreases by 2 per person.
- Additionally, the distance between $A$ and $B$ themselves decreases by 2.

4. **Decrease in $S$:**
- Even if no one is in the interval $(a, b)$, the distance between $A$ and $B$ decreases by 2. Therefore, $S$ decreases by at least 2 per move.
- Since $S$ is a non-negative integer and decreases by at least 2 per move, the process must terminate after a finite number of moves.

5. Final Configuration:
- The final configuration must have all people standing at the same position, as any other configuration would allow further moves.
- Given the invariant that the average position is $1011$, the only possible final configuration is that all people stand at position $1011$.

$\blacksquare$

The final answer is all people standing at position $\boxed{ 1011 }$.