  Based on a recently proposed reinterpretation of gravitational wave memory
that builds up on the definition of gravitational waves pioneered by Isaacson,
we provide a unifying framework to derive both ordinary and null memory from a
single well-defined equation at leading order in the asymptotic expansion. This
allows us to formulate a memory equation that is valid for any unbound
asymptotic energy-flux that preserves local Lorentz invariance. Using Horndeski
gravity as a concrete example metric theory with an additional potentially
massive scalar degree of freedom in the gravitational sector, the general
memory formula is put into practice by presenting the first account of the
memory correction sourced by the emission of massive field waves. Throughout
the work, physical degrees of freedom are identified by constructing manifestly
gauge invariant perturbation variables within an SVT decomposition on top of
the asymptotic Minkowski background, which will in particular prove useful in
future studies of gravitational wave memory within vector tensor theories.
