The Blazhko phenomenon, the modulation of the pulsation of RR Lyrae stars, remains one of the most stubborn unsolved problems of stellar pulsation. The recent idea of Stothers proposes that periodic variations in the properties of the convective envelope may be behind the amplitude and phase modulation. In this work, we approximated the mechanism by introducing variations in the convective parameters of the Florida–Budapest hydrodynamic code and also by means of amplitude equations. We found that the process is only effective for long modulation periods, typically for more than 100 d, in agreement with the thermal time-scales of the pulsation in RR Lyrae stars. Due to the slow response of the pulsation to the structure changes, short-period, high-amplitude Blazhko modulation cannot be reproduced with this mechanism or would require implausible variations in the convective parameters on short time-scales. We also found that the modulation of the mixing length results in strong differences between both the luminosity and radius variations and the respective phase modulations of the two quantities, suggesting notable differences between the energy output of the photosphere and the mechanical variations of the layers. The findings suggest that the convective cycle model is not well suited as a stand-alone mechanism behind the Blazhko effect.