Hint:: If I understand your question correctly -
If your list is $ ( a_1, a_2, \ldots a_n ) $, and your permutation $ \sigma$ must be a permutation of $ ( 1, 2, \ldots, n )$, then the minimum number of moves to go from list to $\sigma$ is
$$ \sum_i | a_i - \sigma(i) | $$
Hence, we are looking for the minimum possible value of this summation, over all possible permutations.
Hint: The minimum occurs when the order of the elements $\sigma^*(i)$ is the exact same as the order of the elements $a_i$. (Note that this is not the only possible equality case.)
You can proof this using the triangle inequality, and consider what happens when you swap $\sigma(i), \sigma(j)$ (i.e. a smoothing argument).