Gap junction turnover
This is an interesting question.
Reading this two articles linked below, it seems very much that the connexins are turned over and degraded with the channel more or less staying at the same place (the figure is taken from the first paper, but the second has a similar):



Following this, the connexins are synthesized and then subjected into the ER and then the Golgi for proper protein maturation and oligomerization into hemichannels (misfold proteins are subjected to proteasomal degradation). They are then exported from the trans-Golgi in small vesicles to the cell membrane, transported to the sarcolemma where they form complete gap junction channes with hemichannels from another cell. Single channels aggregate into gap junction plaques.

The connexins are degraded in the proteasome and the lysosome. It is discussed that post-translational modifications as phosphorylation and ubiquitination play an important role here.

The degradation seems to take place at the formed channels. This was tested in two ways: When the export of new proteins to the membrane is inhibited, then the connexins in the membrane and the junctions get lost over time. If the proteasome is inhibited, the number is growing. And when both pathways are inhibited (export and degradation) the number is approximately constant. See paper 3 for more details. 
This shows that the process of forming the junctions and degrading the connexins is a permanent and that cells, which form junction with each other stay in the same place.

References:


Connexin Expression and Turnover
Pathways for degradation of connexins and gap junctions
Degradation of connexin43 gap junctions involves both the proteasome
and the lysosome.