Topological spin textures can be found in both two-dimensional and three-dimensional nanostructures, which are of great importance to advanced spintronic applications. Here we report the current-induced skyrmion tube dynamics in three-dimensional synthetic antiferromagnetic (SyAF) bilayer and multilayer nanostructures. It is found that the SyAF skyrmion tube made of thinner sublayer skyrmions is more stable during its motion, which ensures that a higher speed of the skyrmion tube can be reached effectively at larger driving current. In the SyAF multilayer with a given total thickness, the current-induced deformation of the SyAF skyrmion tube decreases with an increasing number of interfaces; namely, the rigidity of the SyAF skyrmion tube with a given thickness increases with the number of ferromagnetic (FM) layers. For the SyAF multilayer with an even number of FM layers, the skyrmion Hall effect can be eliminated when the thicknesses of all FM layers are identical. Larger damping parameter leads to smaller deformation and slower speed of the SyAF skyrmion tube. Larger fieldlike torque leads to larger deformation and a higher speed of the SyAF skyrmion tube. Our results are useful for understanding the dynamic behaviors of three-dimensional topological spin textures and may provide guidelines for building SyAF spintronic devices.