BL Lacertae has been the target of several observing campaigns by the Whole Earth Blazar Telescope (WEBT) collaboration and is one of the best studied blazars at all accessible wavelengths. A recent analysis of the optical and radio variability indicates that part of the radio variability is correlated with the optical light curve. Here we present an analysis of a huge VLBI data set including 108 images at 15, 22, and 43 GHz obtained between 1995 and 2004. The aim of this study is to identify the different components contributing to the single-dish radio light curves. We obtain separate radio light curves for the VLBI core and jet and show that the radio spectral index of single-dish observations can be used to trace the core variability. Cross-correlation of the radio spectral index with the optical light curve indicates that the optical variations lead the radio by about 100 days at 15 GHz. By fitting the radio time lags vs. frequency, we find that the power law is steeper than expected for a freely expanding conical jet in equipartition with energy density decreasing as the square of the distance down the jet as in the Königl model. The analysis of the historical data back to 1968 reveals that during a time range of 16 years the optical variability was reduced and its correlation with the radio emission was suppressed. There is a section of the compact radio jet where the emission is weak such that flares propagating down the jet are bright first in the core region with a secondary increase in flux about 1.0 mas from the core. This illustrates the importance of direct imaging to the interpretation of multi-wavelength light curves that can be affected by several distinct components at any given time. We discuss how the complex behaviour of the light curves and correlations can be understood within the framework of a precessing helical jet model.