I was discussing this with my brother. I'm pretty sure I read somewhere that they can move.
EDIT: By movement I mean long distance migration (preferably within the brain only).
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The question is relatively broad and one should take into account that the brain not only consists of neurons, but also glial cells (supportive cells) and pre-mitotic neuronal stem cells. Furthermore, as critical fellow-scientists have indicated, developmental stage is very important, as the developing embryonic brain is very different from the adult brain.
However, after sifting through various publications, the answer to the question is actually remarkably simple: Yes, brain cells migrate.
In the adult brain glial cells migrate in the brain (Klämbt, 2009). Glial cells are involved in a myriad of functions, but a notable example of migrating glial cells are the oligodendrocytes that migrate relative long distances to find their target axons onto which they wrap themselves to form the insulating myelin sheath (Tsai and Miller, 2002).
Neuronal stem cells migrate over long distances in response to injury (Imitola et al., 2004) and they migrate from specific stem-cell locations (e.g., hippocampus and subventricular zone) to other regions (Clarke, 2003).
Interestingly, post-mitotic (fully differentiated, not-dividing) neurons can also be migratory in the adult brain. Specifically, newly formed neurons can migrate over hundreds of microns in fish (Scott et al., 2012) and neuronal migration has been shown in the human cortex as well (Fox et al., 1998).
Not surprisingly, glial cells, stem cells and neurons also migrate during embryonic development. Most notably, post-mitotic neurons destined to fulfill peripheral functions have to migrate over relatively long distances from the neural crest to their target locations (Neuroscience, 2nd ed, Neuronal Migration).
To add to Christiaan's answer, I'll mention one striking example of long-distance neuronal migration in the adult mammalian brain: the so-called Rostral Migratory Stream found in rodents, in rabbits and both the squirrel and rhesus monkey.
Neuronal precursors originating in the subventricular zone (SVZ) of the brain migrate to reach the main olfactory bulb (OB), and it's quite a distance for the small cells.
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