  This article is concerned with two inverse problems on determining moving
source profile functions in evolution equations with a derivative order
$\alpha\in(0,2]$ in time. In the first problem, the sources are supposed to
move along known straight lines, and we suitably choose partial interior
observation data in finite time. Reducing the problems to the determination of
initial values, we prove the unique determination of one and two moving source
profiles for $0<\alpha\le1$ and $1<\alpha\le2$, respectively. In the second
problem, the orbits of moving sources are assumed to be known, and we consider
the full lateral Cauchy data. At the cost of infinite observation time, we
prove the unique determination of one moving source profile by constructing
test functions.
