The prediction of future obesity patterns is crucial for effective strategic planning. However, disproportionally changing body mass index (BMI) distributions pose particular challenges. Flexible modelling of the shape of BMI distributions may improve prediction performance. We used data from repeated national health surveys conducted in Mexico, Colombia and Peru at four or five time points between 1988 and 2014. Data from all surveys except the last survey were used to construct prediction models for three obesity indicators (median BMI, overweight/obesity prevalence and obesity prevalence) for the time of the last survey. We assessed their performance using predicted curves, absolute prediction errors and comparison of actual and predicted distributions. With one method, we modelled the shape of BMI distributions assuming BMI follows a Box-Cox Power Exponential (BCPE) distribution, whose parameters were modelled as a function of interval or nominal 5-year age groups, time and their interaction terms. In a second method, we modelled each of the obesity indicators directly as a function of the same covariates using quantile and logistic regression. The BCPE model with interval age groups yielded the best prediction performance in predicting obesity prevalence. Average absolute prediction errors across all age groups were 4.3 percentage points (95% percentile interval: 1.9, 7.5), 2.5 (1.2, 6.1) and 1.7 (1.0, 9.3), with data from Mexico, Colombia and Peru, respectively. This superiority was weak or none for overweight/obesity prevalence and median BMI. The BCPE model performed better for prediction of the extremes of BMI distribution, possibly by incorporating its shape more precisely. © The Author(s) 2019; all rights reserved. Published by Oxford University Press on behalf of the International Epidemiological Association.