Rotation inside convex Kakeya sets

Let $K$ be a convex body (a compact convex set) in $\mathbb{R}^d$, that contains a copy of another body $S$ in every possible orientation. Is it always possible to continuously move any one copy of $S$ into another, inside $K$? As a stronger question, is it always possible to continuously select, for each orientation, one copy of $S$ in that orientation? These questions were asked by Croft. We show that, in two dimensions, the stronger question always has an affirmative answer. We also show that in three dimensions the answer is negative, even for the case when $S$ is a line segment -- but that in any dimension the first question has a positive answer when $S$ is a line segment. And we prove that, surprisingly, the answer to the first question is negative in dimension four for general $S$.