You can find a lot of such examples in the paper of Abouzaid-Auroux-Katzarkov: https://link.springer.com/article/10.1007/s10240-016-0081-9.

Basically, they studied the case when $X$ is $(\mathbb{C}^\times)^{n-1}\times\mathbb{C}$, and $Y\subset X$ is codimension 2 and is a hypersurface in $(\mathbb{C}^\times)^{n-1}$.

For Fukaya categories, first one should expect a fully faithful embedding

$\Phi_\mathbb{L}:\mathcal{F}(Y)\hookrightarrow\mathcal{F}(Z).$

Such an embedding has already been proved in the case when $X$ is a smooth even-dimensional quadric, and $Y\subset X$ is the base locus of a pencil which includes $X$ as one of its members. The functor $\Phi_\mathbb{L}$ should arise from a Lagrangian correspondence $\mathbb{L}\subset Y^-\times Z$. In the simplest case when $X=\mathbb{C}^2$ and $Y$ is a point, $\mathbb{L}$ is simply the Clifford torus $T\subset\mathbb{C}^2$, which arises as the composition of two correspondences $\mathbb{L}_1\subset\mathit{pt}\times E$ and $\mathbb{L}_2\subset E^-\times Z$. In this case, $\mathbb{L}_1$ is the large circle in the exceptional divisor $E\cong\mathbb{P}^1$, and $\mathbb{L}_2$ is the standard $S^3$ in $Z$, namely the boundary of the unit disc bundle associated to $\mathcal{O}(-1)$. The construction in the pencil of quadric case is just a family version of this. In general, the Lagrangian correspondence $\mathbb{L}_2\rightarrow E$ arises as a bundle over the exceptional divisor $E\subset Z$, whose fibers are Clifford tori in the projective space $\mathbb{P}^k$. You can find constructions of similar flavor in the paper of Abouzaid-Auroux-Katzarkov, in which case you will be forced to deal with non-compact Lagrangian correspondences.

It is expected by many people that the Fukaya category is well-behaved under birational maps, see the introduction of https://arxiv.org/abs/1508.01573. The main result of the paper referred above deals with the case when the center is trivial, namely $Y$ is a point. For related works in the case of Landau-Ginzburg models, see the paper of Kerr: https://arxiv.org/abs/1603.08074.

There are many difficulties in this direction. For example, when the center is not trivial, then you don't get a toric neighborhood in $Z$ after blowing up $Y$, and it's not known in general how to show that the blow-up creates new non-displaceable Lagrangian tori. In the toric case, of course, the existence problem reduces to the analysis of broken holomorphic discs, and this has been done in the paper of Charest-Woodward. In this particular case when the center is trivial, the understanding of $\mathcal{F}(Z)$ reduces essentially to the understanding of $\mathcal{F}(X)$ and $\mathcal{F}(\mathcal{O}(-1))$. The latter one is well-understood by the work of Ritter-Smith.