In a recent paper, A. Fraser and R. Schoen have proved the existence of free boundary minimal surfaces in $B^3$ which have genus $0$ and $n$ boundary components, for all $n > 3$. For large $n$, we give a construction of such surfaces that can be understood as the connected sum of two nearby parallel horizontal discs joined by $n$ boundary bridges which are close to scaled down copies of half catenoids, which are arranged periodically along the unit horizontal great circle of $\mathbb{S}^2$. Furthermore, as $n$ tends to infinity, these free boundary minimal surfaces converge on compact subsets of $B^3$ to the horizontal unit disk taken with multiplicity two.