Intermediate-luminosity red transients (ILRTs) are a class of observed transient posited to arise from the production of an electron-capture supernova from a super-asymptotic giant branch star within a dusty cocoon. In this paper, we present a systematic analysis of narrow Na i D absorption as a means of probing the circumstellar environment of these events. We find a wide diversity of evolution in ILRTs in terms of line strength, time-scale, and shape. We present a simple toy model designed to predict this evolution as arising from ejecta from a central supernova passing through a circumstellar environment wherein Na ii is recombining to Na i over time. We find that while our toy model can qualitatively explain the evolution of a number of ILRTs, the majority of our sample undergoes evolution more complex than predicted. The success of using the Na i D doublet as a diagnostic tool for studying circumstellar material will rely on the availability of regular high-resolution spectral observations of multiple ILRTs, and more detailed spectral modelling will be required to produce models capable of explaining the diverse range of behaviours exhibited by ILRTs. In addition, the strength of the Na i D absorption feature has been used as a means of estimating the extinction of sources, and we suggest that the variability visible in ILRTs would prevent such methods from being used for this class of transient, and any others showing evidence of variability.