We can use demographic methods to study the spatial response of individuals and populations to current global changes. The first mechanism underlying range shifts is a change in the spatial distribution of births and deaths. The spatial regression of demographic rates with geostatistical and spatially explicit models documents the intrinsic growth rate across the range of a population. The population distribution is expected to shift towards areas with the largest intrinsic growth rate, both mechanistically and because these areas are attractive to dispersing individuals. The second mechanism is indeed movement, including emigration away from places that recently became inhospitable and immigration into newly available locations. The analysis of dispersal fluxes using movement data, or indirectly by comparing the observed and intrinsic growth rates in integrated population models, documents these fluxes. Combining these two mechanisms in integral projection models or in individual-based simulations is expected to yield major advances in predictive spatial ecology, that is, mechanistic species distribution models.