  Electron scattering provides a powerful tool to determine charge
distributions and transition densities of nuclei. This tool will soon be
available for short-lived neutron-rich nuclei. [Purpose] Beyond mean-field
methods have been successfully applied to the study of excitation spectra of
nuclei in the whole nuclear chart. These methods permit to determine energies
and transition probabilities starting from an effective in-medium
nucleon-nucleon interaction but without other phenomenological ingredients.
Such a method has recently been extended to calculate the charge density of
nuclei deformed at the mean-field level of approximation [J. M. Yao et al.,
Phys. Rev. C86, 014310 (2012)]. The aim of this work is to further extend the
method to the determination of transition densities between low-lying excited
states. [Method] The starting point of our method is a set of
Hartree-Fock-Bogoliubov wave functions generated with a constraint on the axial
quadrupole moment and using a Skyrme energy density functional. Correlations
beyond the mean field are introduced by projecting mean-field wave functions on
angular-momentum and particle number and by mixing the symmetry restored wave
functions.[Results] We give in this paper detailed formulae derived for the
calculation of densities and form factors. These formulae are rather easy to
obtain when both initial and final states are $0^+$ states but are far from
being trivial when one of the states has a finite $J$-value. Illustrative
applications to $^{24}$Mg and to the even-mass $^{58-68}$Ni have permitted to
analyse the main features of our method, in particular the effect of
deformation on densities and form factors. An illustration calculation of both
elastic and inelastic scattering form factors is presented....
