The calculations performed  are reasonably complex.  The following calculation occurs for vessel 'A' for the downstream vessel 'B' where

R = resistance
compA = compliance of vessel A
capA = capacity of vessel A
volA1 = initial volume of vessel A
volA2 = final volume of vessel A when pressures in both vessels equalised
(Same for vessel B)

The volume that would be transferred if the two vessels were allowed to run to equilibrium is 'volEq' and is given by :

(1)    volEq =[(volA1-capA)*compB-(volB1-capB)*compA]/(compA+compB)

You can derive this yourself or see my derivation

The volume that would be transferred from time zero to time 't' is given by:

(2)     volTransferred=volEq*( 1-exp(-t*K) )

Where K is the rate constant that is the inverse of T the time constant where T = the time taken to reach the equilibrium value if the original rate of flow were maintained.

The intitial rate of flow can be derived from the pressure difference and resistance

(3)     flowRate    = ((volA1-capA)/compA-(volB1-capB)/compB)/R

(4)                 = (volA1-capA)*compB - (volB1-capB)*compA
                                CompA*compB*R

Combining (1) and (4) gives

    (5)     T = volEq/flowRate

           T = [(volA1-capA)*compB -(volB1-capB)*compA] * compA*compB * R
               (compA+compB)* [(volA1-capA)*compB - (volB1-capB)*compA]

     (6)    T = compA*compB * R
                 (compA+compB)

So K = (compA+compB)/(compA*compB * R)