  We consider a discrete system of $n$ devices lying on a 2-dimensional square
grid and forming an initial connected shape $S_I$. Each device is equipped with
a linear-strength mechanism which enables it to move a whole line of
consecutive devices in a single time-step. We study the problem of transforming
$S_I$ into a given connected target shape $S_F$ of the same number of devices,
via a finite sequence of \emph{line moves}. Our focus is on designing
\emph{centralised} transformations aiming at \emph{minimising the total number
of moves} subject to the constraint of \emph{preserving connectivity} of the
shape throughout the course of the transformation. We first give very fast
connectivity-preserving transformations for the case in which the
\emph{associated graphs} of $ S_I $ and $ S_F $ are isomorphic to a Hamiltonian
line. In particular, our transformations make $ O(n \log n $) moves, which is
asymptotically equal to the best known running time of connectivity-breaking
transformations. Our most general result is then a connectivity-preserving
\emph{universal transformation} that can transform any initial connected shape
$ S_I $ into any target connected shape $ S_F $, through a sequence of
$O(n\sqrt{n})$ moves. Finally, we establish $\Omega(n \log n)$ lower bounds for
two restricted sets of transformations. These are the first lower bounds for
this model and are matching the best known $ O(n \log n) $ upper bounds.
