Consider a surface $Σ$ with punctures that serve as marked points and at least one marked point on each boundary component. We build a filling surface $Σ_n$ by singling out one of the boundary components and denoting by $n$ the number of marked points it contains. We consider the triangulations of $Σ_n$ whose vertices are the marked points and the associated flip-graph $\mathcal{F}(Σ_n)$. Quotienting $\mathcal{F}(Σ_n)$ by the homeomorphisms of $Σ$ that fix the privileged boundary component results in a finite graph $\mathcal{MF}(Σ_n)$. Bounds on the diameter of $\mathcal{MF}(Σ_n)$ are available when $Σ$ is orientable and we provide corresponding bounds when $Σ$ is non-orientable. We show that the diameter of this graph grows at least like $5n/2$ and at most like $4n$ as $n$ goes to infinity. If $Σ$ is an unpunctured Möbius strip, $\mathcal{MF}(Σ_n)$ coincides with $\mathcal{F}(Σ_n)$ and we prove that the diameter of this graph grows exactly like $5n/2$ as $n$ goes to infinity.