  This paper establishes non-uniform continuity of the data-to-solution map in
the periodic case, for the two-component Fornberg-Whitham system in Besov
spaces $B^s_{p,r}(\mathbb{T}) \times B^{s-1}_{p,r}(\mathbb{T})$ for $s>
\max\{2+\frac{1}{p}, \frac{5}{2}\}$. In particular, when $p=2$ and $r=2$, this
proves the non-uniform dependence on initial data for the system in Sobolev
spaces $H^s(\mathbb{T})\times H^{s-1}(\mathbb{T})$ for $s> \frac{5}{2}$.
