  We present a search for new planetary-mass members of nearby young moving
groups (YMGs) using astrometry for 694 T and Y dwarfs, including 447 objects
with parallaxes, mostly produced by recent large parallax programs from UKIRT
and Spitzer. Using the BANYAN $\Sigma$ and LACEwING algorithms, we identify 30
new candidate YMG members, with spectral types of T0$-$T9 and distances of
$10-43$ pc. Some candidates have unusually red colors and/or faint absolute
magnitudes compared to field dwarfs with similar spectral types, providing
supporting evidence for their youth, including 4 early-T dwarfs. We establish
one of these, the variable T1.5 dwarf 2MASS J21392676$+$0220226, as a new
planetary-mass member ($14.6^{+3.2}_{-1.6}$ M$_{\rm Jup}$) of the Carina-Near
group ($200\pm50$ Myr) based on its full six-dimensional kinematics, including
a new parallax measurement from CFHT. The high-amplitude variability of this
object is suggestive of a young age, given the coexistence of variability and
youth seen in previously known YMG T dwarfs. Our four latest-type (T8$-$T9) YMG
candidates, WISE J031624.35$+$430709.1, ULAS J130217.21$+$130851.2, WISEPC
J225540.74$-$311841.8, and WISE J233226.49$-$432510.6, if confirmed, will be
the first free-floating planets ($\approx2-6$ M$_{\rm Jup}$) whose ages and
luminosities are compatible with both hot-start and cold-start evolutionary
models, and thus overlap the properties of the directly-imaged planet 51 Eri b.
Several of our early/mid-T candidates have peculiar near-infrared spectra,
indicative of heterogenous photospheres or unresolved binarity. Radial velocity
measurements needed for final membership assessment for most of our candidates
await upcoming 20$-$30 meter class telescopes. In addition, we compile all 15
known T7$-$Y1 benchmarks and derive a homogeneous set of their effective
temperatures, surface gravities, radii, and masses.
