We introduce a class of recursive subhomogeneous algebras which are constructed using a type of diagonal map similar to those previously defined for homogeneous algebras. We call these diagonal subhomogeneous (DSH) algebras. Using homomorphisms that also exhibit a kind of diagonal structure, we study certain limits of DSH algebras. We show that a simple limit of DSH algebras with diagonal maps has stable rank one. As an application we show that whenever $X$ is a compact Hausdorff space and $\sigma$ is a minimal homeomorphism thereof, the crossed product algebra $C^*(\mathbb{Z},X,\sigma)$ has stable rank one. We also define mean dimension in the context of these limits and show that a simple separable limit with mean dimension zero is $\mathcal{Z}$-stable. We also show that the tensor product of any two simple separable limit algebras of this kind is $\mathcal{Z}$-stable.