An earlier version of this post was probably a bit too optimistic and too general.
Let's restrict attention to the example in the question statement: assume the $\mathcal{C}_n$'s are indexed by $\omega^\mathrm{op}$ and are all locally finitely presentable. Then I claim that $\mathcal{C}$ is locally finitely presentable.
Note that we have inclusion functors $\iota_i: \mathcal{C}_i \to \mathcal{C}$ where $(\iota_i \mathcal{C})_n = \begin{cases} F_n \cdots F_{i-1} C & n \leq i \\ 0 & i > n \end{cases}$ where $0$ is the initial object of $\mathcal{C}_n$. The functor $\iota_i$ is left adjoint to the $i$th projection: $\mathcal{C}(\iota_i C, C') = \mathcal{C}_i(C,(C')_i)$, and is fully faithful. Let $\mathcal{K}_i$ ambiguously denote the category of finitely presentable objects in $\mathcal{C}_i$, or its image under $\iota_i$, and write $\iota_i: \mathcal{K}_i \to \mathcal{C}$ for the inclusion. Let $\mathcal{K}$ be the union of the $\mathcal{K}_i$, and write $\iota: \mathcal{K} \to \mathcal{C}$ for the inclusion. It's easy to see that the objects of $\mathcal{K}$ are finitely presentable in $\mathcal{C}$. And they form a dense generator of $\mathcal{C}$:
$[\mathcal{K}^\mathrm{op},\mathsf{Set}](\mathcal{C}(\iota-,C),\mathcal{C}(\iota-,C')) = [\varinjlim_i \mathcal{K_i}^\mathrm{op},\mathsf{Set}](\mathcal{C}(\iota-,C),\mathcal{C}(\iota-,C')) \\ = \varprojlim_i [\mathcal{K_i}^\mathrm{op},\mathsf{Set}](\mathcal{C}(\iota_i-,C),\mathcal{C}(\iota_i-,C')) \\ = \varprojlim_i [\mathcal{K_i}^\mathrm{op},\mathsf{Set}](\mathcal{C}_i(-,C_i),\mathcal{C}_i(-,C'_i)) \\ = \varprojlim_i\mathcal{C}_i(C_i,C'_i) \\ = \mathcal{C}(C,C') \\\\$
where the penultimate step uses that $\mathcal{K}_i$ is dense in $\mathcal{C}_i$.