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 $B_{p,r}^s(\mathbb{T})\times B_{p,r}^{s-1}(\mathbb{T})$ 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 $s>\frac{5}{2}$.