Branching processes are models used to describe populations that reproduce and die over time. In the classical setting, an individual's reproductive capacity remains constant throughout its lifetime. However, in real-world situations, reproductive capacity typically undergoes ageing - that is, after reaching a peak, it decreases over time. In this work, we study the influence of ageing on the behaviour of the process and how modifying its parameters, along with reproduction rates, affects the destiny of the process. More precisely, we introduce an ageing mechanism through a time-dependent birth rate, focusing on the case of exponential decay governed by a parameter $\alpha$. This modification allows us to capture realistic biological and epidemiological scenarios in which reproduction or transmissibility is strongest at early stages and progressively weakens over time. We analyse how the interplay between the reproduction intensity $\lambda$, the ageing parameter $\alpha$, and the spatial structure of the model determines survival and extinction.