  Recently, we have demonstrated a method to record the longitudinal
phase-space distribution of a pulsed cryogenic buffer gas cooled beam of barium
fluoride molecules. In this paper, we use this method to determine the
influence of various source parameters. Besides the expected dependence on
temperature and pressure, the forward velocity of the molecules is strongly
correlated with the time they exit the cell, revealing the dynamics of the gas
inside the cell. Three observations are particularly noteworthy: (1) The
velocity of the barium fluoride molecules increases rapidly as a function of
time, reaches a maximum 50-200 $\mu$s after the ablation pulse and then
decreases exponentially. We attribute this to the buffer gas being heated up by
the plume of hot atoms released from the target by the ablation pulse and
subsequently being cooled down via conduction to the cell walls. (2) The time
constant associated with the exponentially decreasing temperature increases
when the source is used for a longer period of time, which we attribute to the
formation of a layer of isolating dust on the walls of the cell. By thoroughly
cleaning the cell, the time constant is reset to its initial value. (3) The
velocity of the molecules at the trailing end of the molecular pulse depends on
the length of the cell. For short cells, the velocity is significantly higher
than expected from the sudden freeze model. We attribute this to the target
remaining warm over the duration of the molecular pulse giving rise to a
temperature gradient within the cell. Our observations will help to optimize
the source parameters for producing the most intense molecular beam at the
target velocity.
