The BLAKE paper uses round constants derived from $\pi$ to improve diffusion, even if the additions modulo $2^{32}$ are replaced with bitwise XOR when processing a null message and null initial state:
The higher weight in the original model is due to the addition carries induced by the constants $c_0,\dots,c_{15}$. A technique to avoid carries at the first round and get a low-weight output difference is to choose a message such that $m_0 = c_0,\dots,m_{15} = c_{15}$. At the subsequent rounds, however, nonzero words are introduced because of the different permutations.
However, the BLAKE2 paper describes changes it made to BLAKE that remove these constants:
Constants in G initially aimed to guarantee early propagation of carries, but it turned out that the benefits (if any) are not worth the performance penalty. This change saves two xors and two loads per G, that is, 16% of the total arithmetic (addition and xor) instructions.
This statement was made as part of an argument that reducing the number of constants in BLAKE2 does not weaken it, while simultaneously decreasing RAM and ROM footprint. Unfortunately, unlike many of the other trade-offs it discusses, no further explanation is given. What are the benefits of guaranteeing early propagation of carries that lead to BLAKE's constants, and why are these benefits in doubt?