Does using index selectable muxes/demuxes "disable" unselected code for performance reasons?

Hello, this is my first post and I have a question about performance. I'm using the ADAU1701.

If you use index selectable multiplexers and demultiplexers to select between different algorithms, GPIO inputs, or whatever, do the blocks that are out of the signal path still run every sample? Or are those paths paused until selected?

Consider this example: If you select the "Param EQ1" block with GPIO 2, are "Compressor1" and "SoftClip1" still being computed every sample?

A related follow-up question: Is there any advantage to having a "lean" schematic with few blocks, compared to a complicated schematic with lots of blocks? I'm referring to matters such as current consumption and DSP lifespan, not code readability or project maintainability.