Branching stability is a recent concept in point processes and describe the limiting regime in superposition of point processes where particles are allowed to evolve independently according to a subcritical branching process. It is a far-reaching generalisation of the F-stability for non-negative integer random variables introduced in 2004 by Steutel and Van Harn. We fully characterise such processes in terms of their generating functionals and give their cluster representation for the case of non-migrating particles which correspond to Steutel and Van Harn case. We then extend our results to particular important examples of migration mechanism of the particles and characterise the corresponding stability. Branching stable point processes are believed to be an adequate model for contemporary telecommunications systems which show spatial burstiness, like the position of mobile telephones during festival activities in a big city.