The automorphism group of the Poincare' disk has elements called elliptic, which have a single fixed point in the interior of the disk, and can be represented as a rotation around this fixed point.
It also has parabolic elements, which have a single fixed point in the boundary, and in a sense are a rotation around this ideal point.
If $x \in D \to p \in \partial D$, can we see a rotation around $x$ continuosly becoming a rotation around $p$, in some appropriate topology?
Yes. The group of automorphisms of the poincare disc is the Lie group $PSL_2(\mathbb{R})$, so it has natural topology. To see that the phenomena you consider actually happen, take for example the family of rotations $\begin{pmatrix} cos(t^2) & \sin(t^2) \\ -\sin(t^2) & \cos(t^2) \end{pmatrix}$. This is a family of rotations around the origin in the disc. To get a rotation around a point diverging to the boundary, you need to conjugate it by a family of matrices that take the origin to the boundary. An example for that is $\begin{pmatrix} t & 0 \\ 0 & t^{-1} \end{pmatrix}$ for $t \to 0$. By computing the conjugation of the rotation by this family of boosts, you get the family of elliptic elements $$\begin{pmatrix} cos(t^2) & t^2\sin(t^2) \\ -\sin(t^2)/t^2 & \cos(t^2) \end{pmatrix}$$
which converge to $\begin{pmatrix} 1 & 0 \\ -1 & 1 \end{pmatrix}$. This matrix represent a parabolic transformation.
Note that we needed to decrease the angle of rotation in a a speed which is proportional to the parameter of the parameter of the boost for this construction. If the angle of rotation is constant this family diverge.
