I prefer to think that there is only the PRESENT which leaves a trail of MEMORY. And it is only through MEMORY that we recognize the passing of time.
True, but it's also true that everything you interact with in the present has been manifest in some way through past causalities.
So the present is made up of remnants of the past.
The past doesn't technically exist still, except what is left over of it, including our memories.
Really, though, neither 'past' nor 'present' exist in the sense we think of them; because they are just relative to each other as referential concepts. All that really exists are things themselves. Moments in time are not like places that continue to exist after you leave them so you can revisit them later. So you can't say that the present becomes the past or even that the present exists as a continuously updating frame of existence. It is just a way of referring to things temporally in reference to memories and future projections.
If you had the same shoes yesterday as today, then those shoes have existed and continue to do so. There was not a past with different shoes than the present. It's the same pair of shoes whose existence persists while other things come into go out of existence, such as a sneeze.
The irony is that preservation of the past is so important precisely because it doesn't actually exist. If we would waste loads of energy constantly changing and updating everything, it would not only be a terrible increase in entropy; but it would alienate us from our own memories and thus our ability to gain a sense of deliverance and distance from past suffering.