The scarcity of local plant extinctions following recent climate change has been explained by demographic inertia and lags in the displacement of resident species by novel species, generating an “extinction debt”. We established a transplant experiment to disentangle the contribution of these processes to local extinction risk of four alpine plants in the Swiss Alps. Projected population growth (λ) derived from integral projection models was reduced by 0.07/°C of warming on average, while novel species additionally decreased λ by 0.15 across warming levels. Effects of novel species on predicted extinction time were greatest at warming <2°C for two species. Projected population declines under both warming and with novel species were primarily driven by increased mortality. Our results suggest that extinction debt can be explained by a combination of demographic inertia and lags in novel species establishment, with the latter being particularly important for some species under low levels of warming.