Dispersal, and in particular the frequency of long-distance dispersal (LDD) events, has strong implications for population dynamics with possibly the acceleration of the colonisation front, and for evolution with possibly the conservation of genetic diversity along the colonised domain. However, accurately inferring LDD is challenging as it requires both large-scale data and a methodology that encompasses the redistribution of individuals in time and space. Here, we propose a mechanistic-statistical framework to estimate dispersal from one-dimensional invasions. The mechanistic model takes into account population growth and grasps the diversity in dispersal processes by using either diffusion, leading to a reaction-diffusion (R.D.) formalism, or kernels, leading to an integro-differential (I.D.) formalism. The latter considers different dispersal kernels (e.g. Gaussian, Exponential, and Exponential-power) differing in their frequency of LDD events. The statistical model relies on dedicated observation laws that describe two types of samples, clumped or not. As such, we take into account the variability in both habitat suitability and occupancy perception. We first check the identifiability of the parameters and the confidence in the selection of the dispersal process. We observed good identifiability for all parameters (correlation coefficient >0.9 between true and fitted values). The dispersal process that is the most confidently identified is Exponential-Power (i.e. fat-tailed) kernel. We then applied our framework to data describing an annual invasion of the poplar rust disease along the Durance River valley over nearly 200 km. This spatio-temporal survey consisted of 12 study sites examined at seven time points. We confidently estimated that the dispersal of poplar rust is best described by an Exponential-power kernel with a mean dispersal distance of 1.94 km and an exponent parameter of 0.24 characterising a fat-tailed kernel with frequent LDD events. By considering the whole range of possible dispersal processes our method forms a robust inference framework. It can be employed for a variety of organisms, provided they are monitored in time and space along a one-dimension invasion.