The light that we receive from clusters of galaxies is redshifted by the presence of the clusters' gravitational potential. This effect, known as gravitational redshift, was first detected from a sample of stacked clusters in 2011, by taking redshift differences between the centre of each cluster and the respective member galaxies. However, the interpretation of this result was later challenged by several studies, which emphasised the possible influence of additional kinematic effects on the observed signal, like the transverse Doppler effect. In this work, we present the first derivation of all such effects within a relativistic framework, accurate to third order in the weak-field approximation. This framework allows us to correctly capture the hierarchy of terms on the scale of clusters and at the same time account for all relativistic effects. We compare our result with previous literature and show that some terms of the same order of the transverse Doppler effect were not properly included, leading to an overestimation of the kinematic contamination. In particular, we do not find any contribution arising from the so-called light-cone effect and obtain a larger correction due the motion of the central galaxy. Our derivation is independent of the Euler equation, providing a straightforward framework to test the weak equivalence principle.