Necessary and Sufficient Condition for A Particular Sum Rearrangement

Question

Let $\phi:\mathbb{N}\rightarrow\mathbb{N}$ be a rearrangement of $\mathbb{N}$ (a bijection). I am searching for a condition equivalent to: $$$$ For all complex sequences $(\alpha_n)$, there exists some $l\in\mathbb{N}$ such that for all $N\in\mathbb{N},$ $$\sum_{n=0}^N(\alpha_{\phi_n}-\alpha_n)=O\left(\max_{|n-N|\leq l}|\alpha_n|\right).$$ So far, I've proven that if $(\phi_n-n)=O(1),$ then this condition follows. I then proved that if this condition holds, $\sum_{n=0}^N(\phi_n-n)=O(N)$ follows. Then, heuristically, I imagine there should be some condition, implied by $(\phi_n-n)=O(1),$ and implying $\sum_{n=0}^N(\phi_n-n)=O(N),$ which is equivalent to the condition above (and about as simply stated as each of these two conditions). Does anyone know of such a result? I know there are similar results about convergence-preserving rearrangements, though this condition is stronger in the following sense: $$$$ Such a $\phi$ would have the following properties: $$$$ (i) If $\left(\sum_{n=0}^N\alpha_n\right)$ converges, then $\left(\sum_{n=0}^N\alpha_{\phi_n}\right)$ converges and $\sum_{n=0}^\infty\alpha_{\phi_n}=\sum_{n=0}^\infty\alpha_n.$ $$$$ (ii) If $\left(\sum_{n=0}^N\alpha_n\right)$ is bounded, then $\left(\sum_{n=0}^\infty\alpha_{\phi_n}\right)$ is bounded. $$$$ (iii) If $\left(\sum_{n=0}^N\alpha_n\right)$ is unbounded, then $\left(\sum_{n=0}^\infty\alpha_{\phi_n}\right)$ is unbounded.