Recently it was shown that standard odd- and even-dimensional general relativity can be obtained from a (2n + 1)-dimensional Chern–Simons Lagrangian invariant under the B2n+1 algebra and from a (2n)-dimensional Born–Infeld Lagrangian invariant under a subalgebra LB2n+1, respectively. Very recently, it was shown that the generalized Inönü–Wigner contraction of the generalized AdS–Maxwell algebras provides Maxwell algebras of types Mm which correspond to the so-called Bm Lie algebras. In this article we report on a simple model that suggests a mechanism by which standard odd-dimensional general relativity may emerge as the weak coupling constant limit of a (2p + 1)- dimensional Chern–Simons Lagrangian invariant under the Maxwell algebra type M2m+1, if and only if m ≥ p. Similarly, we show that standard even-dimensional general relativity emerges as the weak coupling constant limit of a (2p)- dimensional Born–Infeld type Lagrangian invariant under a subalgebra LM2m of the Maxwell algebra type, if and only if m ≥ p. It is shown that when m < p this is not possible for a (2p +1)-dimensional Chern–Simons Lagrangian invariant under the M2m+1 and for a (2p)-dimensional Born–Infeld type Lagrangian invariant under the LM2m algebra.