  We investigate the detection prospects for two-neutrino and neutrinoless
second order weak decays of $^{124}$Xe -- double electron capture
($0/2\nu\text{ECEC}$), electron capture with positron emission
($0/2\nu\text{EC}\beta^+$) and double-positron emission
($0/2\nu\beta^+\beta^+$) -- in multi-tonne xenon time projection chambers. We
simulate the decays in a liquid xenon medium and develop a reconstruction
algorithm which uses the multi-particle coincidence in these decays to separate
signal from background. This is used to compute the expected detection
efficiencies as a function of position resolution and energy threshold for
planned experiments. In addition, we consider an exhaustive list of possible
background sources and find that they are either negligible in rate or can be
greatly reduced using our topological reconstruction criteria. In particular,
we draw two conclusions: First, with a half-life of
$T_{1/2}^{2\nu\text{EC}\beta^+} = (1.7 \pm 0.6)\cdot 10^{23}\,\text{yr}$, the
$2\nu\text{EC}\beta^+$ decay of $^{124}$Xe will likely be detected in upcoming
Dark Matter experiments (e.g. XENONnT or LZ), and their major background will
be from gamma rays in the detector construction materials. Second, searches for
the $0\nu\text{EC}\beta^+$ decay mode are likely to be background-free, and new
parameter space may be within the reach. To this end we investigate three
different scenarios of existing experimental constraints on the effective
neutrino mass. The necessary 500 kg-year exposure of $^{124}$Xe could be
achieved by the baseline design of the DARWIN observatory, or by extracting and
using the $^{124}$Xe from the tailings of the nEXO experiment. We demonstrate
how a combination of $^{124}$Xe results with those from $0\nu\beta^-\beta^-$
searches in $^{136}$Xe could help to identify the neutrinoless decay mechanism.
