  We present a comprehensive study of the rotational and emission properties of
PSR J0738$-$4042 using a combination of observations taken by the Deep Space
Network, Hartebeesthoek, Parkes (Murriyang) and Molonglo observatories between
1972 and 2023. Our timing of the pulsar is motivated by previously reported
profile/spin-down events that occurred in September 2005 and December 2015,
which result in an anomalously large braking index of $n = 23300 \pm 1800$.
Using a Gaussian process regression framework, we develop continuous models for
the evolution of the pulsar spin-down rate ($\dot{\nu}$) and profile shape. We
find that the pulse profile variations are similar regardless of radio
observing frequency and polarisation. Small-scale differences can be ascribed
to changes in the interstellar medium along the line of sight and
frequency-dependent changes in magnetospheric radio emission height. No new
correlated spin-down or profile events were identified in our extended dataset.
However, we found that the disappearance of a bright emission component in the
leading edge of archival profiles between 1981-1988 was not associated with a
substantial change in $\dot{\nu}$. This marks a notable departure from the
previous profile/spin-down events in this pulsar. We discuss the challenges
these observations pose for physical models and conclude that interactions
between the pulsar and in-falling asteroids or a form of magnetospheric
state-switching with a long periodicity are plausible explanations.
