Electric and thermal transport properties of a $\nu=2/3$ fractional quantum Hall junction are anaylzed. We investigate the evolution of the electric and thermal two-terminal conductances, $G^{(2)}$ and $G^{\rm th, (2)}$ with system size $L$ and temperature $T$. This is done both for the case of strong interaction between the 1 and 1/ 3 modes (when the low-temperature physics of the interacting segment of the device is controlled by the vicinity of the strong-disorder Kane-Fisher-Polchinski fixed point) and for relatively weak interaction, for which the disorder is irrelevant at $T=0$ in the renormalization-group sense. The transport properties in both cases are similar in several respects. In particular, $G^{(2)}(L)$ is close to 4/3 (in units of $e^2/h$) and $G^{\rm th, (2)}$ to 2 (in units of $\pi T / 6 \hbar$) for small $L$, independently of the interaction strength. For large $L$ the system is in an incoherent regime, with $G^{(2)}$ given by 2/3 and $G^{\rm th, (2)}$ showing the Ohmic scaling, $G^{\rm th, (2)}\propto 1/L$, again for any interaction strength. The hallmark of the strong-disorder fixed point is the emergence of an intermediate range of $L$, in which the electric conductance shows strong mesoscopic fluctuations and the thermal conductance is $G^{\rm th, (2)}=1$. The analysis is extended also to a device with floating 1/3 mode, as studied in a recent experiment [A. Grivnin et al, Phys. Rev. Lett. {\bf 113}, 266803 (2014)].