How does the mass encased within the Hubble sphere/volume (both proper and comoving) change with time?
It seems that we live in a universe which is accelerating, but with a decreasing Hubble constant (See How is the Universe's Expansion Accelerating if the Hubble Constant is Decreasing?)
In this universe the Hubble volume expands with time, and can overtake sources that are currently more distant than the Hubble length. That is the Hubble length increases faster than those objects are receding. And so the mass in the Hubble volume will increase.