  In a reconfiguration version of an optimization problem $\mathcal{Q}$ the
input is an instance of $\mathcal{Q}$ and two feasible solutions $S$ and $T$.
The objective is to determine whether there exists a step-by-step
transformation between $S$ and $T$ such that all intermediate steps also
constitute feasible solutions. In this work, we study the parameterized
complexity of the \textsc{Connected Dominating Set Reconfiguration} problem
(\textsc{CDS-R)}. It was shown in previous work that the \textsc{Dominating Set
Reconfiguration} problem (\textsc{DS-R}) parameterized by $k$, the maximum
allowed size of a dominating set in a reconfiguration sequence, is
fixed-parameter tractable on all graphs that exclude a biclique $K_{d,d}$ as a
subgraph, for some constant $d \geq 1$. We show that the additional
connectivity constraint makes the problem much harder, namely, that
\textsc{CDS-R} is \textsf{W}$[1]$-hard parameterized by $k+\ell$, the maximum
allowed size of a dominating set plus the length of the reconfiguration
sequence, already on $5$-degenerate graphs. On the positive side, we show that
\textsc{CDS-R} parameterized by $k$ is fixed-parameter tractable, and in fact
admits a polynomial kernel on planar graphs.
