Physicists usually compute dimensions by using boxes and they also do so when dealing with multifractals. Also in the study of some dynamical systems and multiplicative processes, boxes naturally appear. On the other hand, in geometric measure theory, it is preferred to perform computations which do not depend on a grid. This paper provides a bridge between the boxes and the grid-free approaches to the multifractal analysis of measures. Results for quasi-Bernoulli measures and statistically self-similar measures are obtained.