Neurons do move. Not in a sense that they float around as red blood cells do in blood vessel, but in a sense that they make new branches and stretch them to make new synaptic connections or disconnect existing synapses and degenerate branches. The process is called the neuroplasticity and it is the very basis of learning and memory.
There are a few facts to consider about the plasticity in the retina and about your question.
1) In the cubes, there are non-neuronal cells called as glia, but they are also a part of neural system. Sometimes you may also encounter blood vessel with thick swirling wall. But you're right. As you pointed out, the neurons are very tightly packed to each other. On the other hand, the neurons we are looking at were blown up a little bit during the sample preparation, making the extracellular space a little smaller than when it was in the living mouse. Nevertheless, the effect must not be critical.
2) The plasticity in retina is not as strong as that of the rest of the brain. It is believed that the retinal neural network is mainly designed by the genes. Each neuron is told by the genes what to do, where to go, and which other neurons to connect to. An organism processes the visual information in almost the same way as other individuals of its kind do.
3) So let's talk about the plasticity in general. While scientists know the cellular and molecular mechanism of the process pretty well, and they can even observe a few neurons change their shapes and connections on a petri dish, I'm not sure if anyone has ever seen this in a bulk tissue simply because it would be technically difficult. To do this, you will have to find a way to "look through" the bulk and will have to know which neuron to look at.
So to tell the truth, we do not know so much about what really happens during the process of plasticity. Given the circumstances, it wouldn't be too wrong to assume that a growing branch of a neuron must crawl its way through the jungle of other neurons, but the final answer is yet to be studied.