We study several problems on geometric packing and covering with movement. Given a family $mathcal{I}$ of $n$ intervals of $kappa$ distinct lengths, and another interval $B$, can we pack the intervals in $mathcal{I}$ inside $B$ (respectively, cover $B$ by the intervals in $mathcal{I}$) by moving $tau$ intervals and keeping the other $sigma = n - tau$ intervals unmoved? We show that both packing and covering are W[1]-hard with any one of $kappa$, $tau$, and $sigma$ as single parameter, but are FPT with combined parameters $kappa$ and $tau$. We also obtain improved polynomial-time algorithms for packing and covering, including an $O(nlog^2 n)$ time algorithm for covering, when all intervals in $mathcal{I}$ have the same length.
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