There are two different things one can mean by time.
One is purely the time dimension in Space-time. It is a direction just like any spatial direction. The difference is in its metric, but that's a separate discussion.
The second is time as sequence of events. A caused B, B caused C, and so on. This has to do with entropy, information, and is responsible for apparent "flow" of time. This time is governed by our Sun, as it is our primary energy source. But again, that's something completely different.
Matter is said to propagate through time. That doesn't mean movement in the sense you are used to it, because you usually think of movement as change of position with respect to time. How can position in time change with respect to time? It cannot. There is, however, something called proper time.
Any particle in 4-space is not a point. It is a curve in 4-dimensions. We only see one point along that curve at anyone time. Distance along that curve is called particle's proper time. To each "moment" in proper time corresponds one point on the curve. The rate with which that point moves with respect to proper time is called proper velocity. The time component of proper velocity is the rate at which particle propagates through time.
If this is a bit confusing, you might need to take it one step at a time. You might want to take a look at
this Wikipedia article.