A theoretical framework to investigate the time evolution of the quantum entanglement due to the dynamical Lamb effect between $N$ superconducting qubits coupled to a coplanar waveguide in the presence of different sources of dissipation is developed. We quantitatively analyze the case of $N=2$ and $3$ qubits under the assumptions of single switching of the coupling and absence of dissipation within a perturbative approach. The same systems are analyzed for the general case of periodic switching of the coupling in the presence of dissipation via numerical calculations. Different measures of entanglement compatible with mixed states are adopted. It is demonstrated that the different measures show different level of details of the latter. The concurrence and the negativity are obtained in the two qubits case, the three-$pi$ and the negativity in the three qubits case. It is shown that time-dependent Greenberger-Horne-Zeilinger states can be created even in presence of dissipation. To maximize the quantum entanglement between the qubits, the effects of tuning several parameters of the system are investigated.
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