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Dislocations move because the resolved shear stress (Schmidfactor) is sufficient high enough to activate the glide system in which the slip plane lies. (slip)
Screw-dislocations in fact are able to move from one slip plane to another parallel plane (burgers vector does not change!).
In a fcc lattice a $dislocation may change the (111) plane to (1 -1 1) plane due to local stresses. (cross slip)
Edge dislocations can change their planes too. It is a thermally activated process which is called climb (vacancy flux).
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