The dynamic partial sorting problem asks for an algorithm that maintains lists of numbers under the link, cut and change value operations, and queries the sorted sequence of the $k$ least numbers in one of the lists. We first solve the problem in $O(k\log (n))$ time for queries and $O(\log (n))$ time for updates using the tournament tree data structure, where $n$ is the number of elements in the lists. We then introduce a layered tournament tree data structure and solve the same problem in $O(\log_\varphi^* (n) k\log (k))$ time for queries and $O\left(\log (n)\cdot\log^2\log (n)\right)$ for updates, where $\varphi$ is the golden ratio and $\log_\varphi^*(n)$ is the iterated logarithmic function with base $\varphi$.