Connections between certain numbers related to derangements and $r$-permutations

For non-negative integer parameters $r,u,m,n$ define \begin{align*} \cal{D}(r,u,m,n) := \big{\ σ\in \cal{S}{r+n}\ \big|\ σ(x)=y $\text{ for exactly }$ u $\text{ pairs }$ (x,y) $\text{ such that }$ 1\leq x,y\leq r $\text{ and }$ σ(t)=t $\text{ for exactly }$ m $\text{ elements }$ r+1\leq t\leq r+n\ \big} \end{align*} and \begin{align*} \cal{D}{r,u,m}(n) := \big{\ σ\in \cal{S}{r+n}\ \big|\ \forall{1\leq x<y\leq r} \ x $\text{ and }$ y $\text{ are in disjoint cycles of }$ σ$\text{ and }$ σ(z)=z $\text{ for exactly }$ u $\text{ elements }$ 1\leq z\leq r, $\text{ and }$ σ(t)=t $\text{ for exactly }$ m $\text{ elements }$ r+1\leq t\leq r+n\ \big}, \end{align*} where ${\mathcal{S}}_n$ denotes the set of all the permutations of $\{1,\dots ,n\}$. In this paper we study connections between the sets $\mathcal{D}(r,u,m,n)$, ${\mathcal{D}}_{r,u,m}(n)$, and the sets of (some classes of) $r$-derangements. We rely mostly on counting arguments.