The paper presents a general solution of the isotropic elasticity problem for the sandwiched inclusion in an elastic cylinder and having axially varying eigenstrain. This problem is relevant in considerations of a cylinder with axially varying thermal expansion coefficient or a nanowire having an axial inhomogeneity of the crystal lattice parameter due to growth and processing conditions. The technique of the solution and the results for the elastic fields and energies of dilatational inclusions with different distribution of eigenstrain, namely, constant, trapezoidal, and diffusion-like, along the axis of the cylinder, are given in full details. The technique explores the axial superpositions of infinitely thin dilatation disks inserted in the cylinder, the elastic fields of which are found in analytical form. In addition, the energies of interfaces separated domains with constant eigenstrains in the cylinder are given. It is demonstrated that the blurring of the eigenstrain in the transition region leads to a decrease in interface energy.