In this paper, we showthat a small sample of radio galaxies with evidence for multiple epochs of jet activity (so-called double-double radio galaxies) have the same electron injection spectral index in the two activity episodes, a result which might be considered surprising given the very different lobe dynamics expected in the first and second episode. We construct models for the dynamics of radio galaxies, with an emphasis on their episodic behaviour, and show that hotspot formation and confinement of lobes for the inner double of double-double radio galaxies are possible even without any thermal matter in the outer cocoon. We argue that (i) the observed similar injection spectral indices are due to similar jet powers in the two episodes, (ii) the 'spectral index-radio power' correlation of a flux limited sample of radio galaxies is the primary one, and not the 'spectral index-redshift correlation', (iii) jets are made of pair plasma and not electron-proton and (iv) the Lorentz factor of the spine of the jet should be ≳10 to explain the observations. Furthermore, we argue that the observations show that higher power radio galaxies do not have a higher jet bulk Lorentz factors, but instead simply have a higher number density of particles in the jet rest frame. A consequence of our models is that aligned double-double radio galaxies with very old (≳10 yr) outer doubles, or misaligned double-double radio galaxies, are statistically more likely to have dissimilar injection indices in two different episodes, as they will probably have different jet powers.