You are correct in your statement about having to toggle between the two multiplexers to take a proper reading of any single input. You do this by setting or resetting the Enable bit which is included in the SMBus-delivered configuration word ( Page 8 of the Linear Tech data sheet). You must select one or the other of the two multiplexers. So, there are a few precautions.
A. You must avoid enabling both multiplexers at the same time as this will effectively connect two of your inputs together. Since these inputs are sourced by some entitiy, you would effectively be shorting these two sources together. Maybe OK, maybe not, depending on their source impedances and electrical composition. So, best to avoid this situation entirely.
B. When switching between two inputs on opposite multiplexers, always go to the "both off" state first, then select the appropriate multiplexer in a subsequent SMB transaction. (See "C" below, as this is a transient floating condition.) This is essentially a "break-before-make" strategy. Not to worry when you are switching between inputs on a single multiplexer because the LTC1380 has "Guaranteed Break-Before-Make" (Page 1 of data sheet).
C. If you disable both 1380's simultaneously, the NI input of your unity-gain buffer will be left floating - maybe OK, maybe not, depends on a number of factors. Unless you leave both MUX's disabled for an extended period of time ( say more than 100 milli-secs ), I wouldn't worry about it. You could connect a high value resistor ( 1 meg or greater) from the NI input to DC Common to eliminate this floating situation, but that could create accuracy issues, depending on the source impedances of the input signals and a few other factors. As a practical precaution you should probably assign one of the inputs as a "parking" position. Always select this input when you are not actively utilizing any of the other inputs.