In that framework something most still be moving through an external time to be experiencing individual parts of it sequentially.
When you experience a moment of consciousness at a specific 4D position, you experience all the memories of the "past" that are in your brain. Thus, the consciousness particle interpretation of consciousness is compatible with your direct perception of living a 3D life along a time dimension.
[1] https://en.wikipedia.org/wiki/Integrated_information_theory
We are on the inside of a 4 dimensional bulk. We are traveling through the temporal dimension.
As opposed to outside of it looking in. In which case we would "see" past, present, and future simultaneously.