Can I formulate 'value shifting' constraint with MILP?

I am trying to figure out if it is possible for this 'value shifting' constraint to be implemented in MILP.

In summary there is a vector used for other part of the model, and I am thinking of adding a bit of flexibility to the vector.

The idea is to give each vector value the freedom to shift to the right, up to a maximum of a positions, after the optimal shifting, the new vector will then be used in other parts of the optimization model for better minimization.

E.g.

x_old = [1 2 3 4 5] % x(1) goes to x(2), x(3) goes to x(5)
x_new = [0 3 0 4 8] % [0 x(1)+x(2) 0 x(4) x(3)+x(5)]

Boundary case of x(4) and x(5) will loop back to the start, allowing the re-allocation to the start of the vector

The only thing I came up so far is to have an Integer positional vector p with lower bound of 0 and upper bound a, then loop through the original vector and create x_new, but that would involve custom non-linear constraints like

 x_new(ii+p(ii)) = x_new(ii+p(ii)) + x_old(ii)

Thanks!