Long-lived plant species are highly valued environmentally, economically, and socially, but can also cause substantial harm as invaders. Realistic demographic predictions can guide management decisions, and are particularly valuable for long-lived species where population response times can be long. Long-lived species are also challenging, given population dynamics can be affected by factors as diverse as herbivory, climate, and dispersal. We developed a matrix model to evaluate the effects of herbivory by a leaf-feeding biological control agent released in Australia against a long-lived invasive shrub (mesquite, Leguminoseae: Prosopis spp.). The stage-structured, density-dependent model used an annual time step and 10 climatically diverse years of field data. Mesquite population demography is sensitive to source–sink dynamics as most seeds are consumed and redistributed spatially by livestock. In addition, individual mesquite plants, because they are long lived, experience natural climate variation that cycles over decadal scales, as well as anthropogenic climate change. The model therefore explicitly considered the effects of both net dispersal and climate variation. Herbivory strongly regulated mesquite populations through reduced growth and fertility, but additional mortality of older plants will be required to reach management goals within a reasonable time frame. Growth and survival of seeds and seedlings were correlated with daily soil moisture. As a result, population dynamics were sensitive to rainfall scenario, but population response times were typically slow (20–800 years to reach equilibrium or extinction) due to adult longevity. Equilibrium population densities were expected to remain 5% higher, and be more dynamic, if historical multi-decadal climate patterns persist, the effect being dampened by herbivory suppressing seed production irrespective of preceding rainfall. Dense infestations were unlikely to form under a drier climate, and required net dispersal under the current climate. Seed input wasn't required to form dense infestations under a wetter climate. Each factor we considered (ongoing herbivory, changing climate, and source–sink dynamics) has a strong bearing on how this invasive species should be managed, highlighting the need for considering both ecological context (in this case, source–sink dynamics) and the effect of climate variability at relevant temporal scales (daily, multi-decadal, and anthropogenic) when deriving management recommendations for long-lived species.