Birth-death models are used to understand the interplay of genetic drift and natural selection. While well-mixed populations remain unaffected by the order of birth and death and where selection acts, evolutionary outcomes in spatially structured populations are affected by these choices. We show that the choice of individual moving to vacant sites-parent or offspring-controls the initial mutant placement on a graph and hence alters its fixation probability. Moving parent individuals introduces, to our knowledge, previously unexplored update rules and fixation categories for heterogeneous graphs. We identify a class of graphs, amplifiers of fixation, where fixation probability is larger than in well-mixed populations, regardless of the mutant fitness. Under death-Birth parent moving, the star graph is an amplifier of fixation, with a non-zero fixation probability for deleterious mutants, in contrast to very large well-mixed populations. Most Erdős-Rényi graphs of size 8 are amplifiers of fixation under death-Birth parent moving, but suppressors of fixation under Birth-death offspring moving. Surprisingly, amplifiers of fixation attain lower fitness in long-term evolution, despite favouring beneficial mutants, while suppressors of fixation attain higher fitness. These counterintuitive findings are explained by the fate of deleterious mutations and highlight the crucial role of deleterious mutants for adaptive evolution.