Blazars are characterized by flux variability that is frequency-dependent and manifests in a varietyof timescales. Decade-long monitoring of blazars at optical and infrared (OIR) wavelengthswith the Small and Moderate Aperture Research Telescope System (SMARTS) in Chile and ingamma-rays with the Fermi Large Area Telescope (LAT) has enabled the systematic study of theirmulti-wavelength (MW) long-term variability. These studies pinpoint that besides extreme short-term variability (minutes to hours), a source can exhibit correlated and uncorrelated inter-band fluxvariability between different observation periods and/or on different timescales (days to years).In this work, we investigate from a theoretical perspective the long-term variability properties ofblazar emission. To do so, we impose variations on the main parameters of the one-zone leptonicmodel (injection luminosity of relativistic electrons, strength of magnetic field, Doppler factor,and external photon field luminosity) motivated by the Fermi-LAT full-mission light curves of blazars. Using as case studies two bright and well-monitored blazars from the SMARTS sample(PKS2155-304 and 3C273), we compute 10 year-long OIR, X-ray, and gamma-ray model lightcurves for different varying parameters. We compare the findings of our theoretical investigation with MW observations using various measures of variability. While no single-varying parameter simulation can explain all MW variability properties, our results motivate future time-dependent studies with coupling between two or more physical parameters to describe the MW long-term blazar variability.