Understanding and forecasting human mobility in response to climatic and environmental changes has become a subject of substantial political, societal, and academic interest. Quantitative models exploring the relationship between climatic factors and migration patterns have been developed since the early 2000s; however, different models have produced results that are not always consistent with one another or robust enough to provide actionable insights into future dynamics. Here we examine weaknesses of classical methods and identify next-generation approaches with the potential to close existing knowledge gaps. We propose six priorities for the future of climate mobility modeling: (i) the use of non-linear machine-learning rather than linear methods, (ii) the prioritization of explaining the observed data rather than testing statistical significance of predictors, (iii) the consideration of relevant climate impacts rather than temperature- and precipitation-based metrics, (iv) the examination of heterogeneities, including across space and demographic groups rather than aggregated measures, (v) the investigation of temporal migration dynamics rather than essentially spatial patterns, (vi) the use of better calibration data, including disaggregated and within-country flows. Improving both methods and data to accommodate the high complexity and context-specificity of climate mobility will be crucial for establishing the scientific consensus on historical trends and future projections that has eluded the discipline thus far.