  We study semigroups $(\phi_t)_{t\geq 0}$ of holomorphic self-maps of the unit
disk with Denjoy-Wolff point on the boundary. We show that the orthogonal speed
of such semigroups is a strictly increasing function. This answers a question
raised by F. Bracci, D. Cordella, and M. Kourou, and implies a domain
monotonicity property for orthogonal speeds conjectured by Bracci. We give an
example of a semigroup such that its total speed is not eventually increasing.
We also provide another example of a semigroup having total speed of a certain
asymptotic behavior, thus answering another question of Bracci.
