Heat continues to flow between the tubs through the conductors, from hotter areas to colder ones, until the temperature difference between them is (including the connectors) **zero**, i.e. they are all at the same temperature $T_e$ ([Zeroth Law of thermodynamics][1]): full **thermodynamic equilibrium** has been reached.

Because the system is adiabatic ($\Sigma Q=0$), we can write (initial state v. end state):

$$m_1cT_1+m_2cT_2+m_3cT_3=m_1cT_e+m_2cT_e+m_3cT_e$$

Assuming $m_1c=m_2c=m_3c$ ($m$ is water mass, $c$ water specific heat),

we get simply:

$$T_e=\frac{T_1+T_2+T_3}{3}$$

The nature of the connectors does not affect the end result, only the time needed to reach full thermodynamic equilibrium.


  [1]: https://en.wikipedia.org/wiki/Zeroth_law_of_thermodynamics