  A planar framework -- a graph together with a map of its vertices to the
plane -- is flexible if it allows a continuous deformation preserving the
distances between adjacent vertices. Extending a recent previous result, we
prove that a connected graph with a countable vertex set can be realized as a
flexible framework if and only if it has a so-called NAC-coloring. The tools
developed to prove this result are then applied to frameworks where every
4-cycle is a parallelogram, and countably infinite graphs with $n$-fold
rotational symmetry. With this, we determine a simple combinatorial
characterization that determines whether the 1-skeleton of a Penrose rhombus
tiling with a given set of braced rhombi will have a flexible motion, and also
whether the motion will preserve 5-fold rotational symmetry.
