We introduce a general class of branching Markov processes for the modelling of structured populations. Each individual has a nonnegative trait (size, quantity of parasites or proteins, ...), which evolves as a diffusion with positive jumps. The growth rate, diffusive function and jump rate of this trait may depend on the trait value. The individual death rate also depends on the trait. At death, an individual gives birth to two offspring and its trait is shared (unequally) between these two offspring. We study the long time behaviour of the trait along a lineage and in the whole population.