Studies leveraging gene-environment (GxE) interactions within Mendelian randomization (MR) analyses have prompted the emergence of two similar methodologies: MR-GxE and MR-GENIUS. Such methods are attractive in allowing for pleiotropic bias to be corrected when using individual instruments. Specifically, MR-GxE requires an interaction to be explicitly identified, while MR-GENIUS does not. We critically examine the assumptions of MR-GxE and MR-GENIUS in the absence of a pre-defined covariate, and propose sensitivity analyses to evaluate their performance. Finally, we explore the effect of body mass index (BMI) upon systolic blood pressure (SBP) using data from the UK Biobank, finding evidence of a positive effect of BMI on SBP. We find both approaches share similar assumptions, though differences between the approaches lend themselves to differing research settings. Where a suitable gene-by-covariate interaction is observed MR-GxE can produce unbiased causal effect estimates. MR-GENIUS can circumvent the need to identify interactions, but as a consequence relies on either the MR-GxE assumptions holding globally, or additional information with respect to the distribution of pleiotropic effects in the absence of an explicitly defined interaction covariate.