Salt Bridges/Porous Disks
It depends on the geometry/construction of the bridge. In a typical glass-tube bridge, some small amount of the ions from the cell may eventually make their way all the way across the salt bridge, but over reasonable amounts of time, it's mostly the ions from the salt bridge that move into the half-cells (and the ions from the half-cells moving into the ends of the bridge). For a small filter paper strip being used as a bridge, the time it takes should be shorter, as there is much less solution in the salt bridge and the distance is likely shorter. Sometimes, glass salt bridges are filled with gels to limit diffusion further (at the expense of increasing the resistance of the bridge).

The goal of using a salt bridge and separating the half-cells is to prevent the solutions of each from directly mixing. If significant quantities of the ions from the half-cells are crossing the bridge, then they can react directly, bypassing the electrode connections. e.g. if we have a simple galvanic cell of $\ce{Zn∣Zn^2+∥Cu^2+∣Cu}$ and $\ce{Cu^2+}$ makes it across the separator to the zinc size, it can be directly reduced onto the zinc electrode, instead of by the electrons flowing through the electrode connections. Having a single beaker with both $\ce{Cu^2+}$ and $\ce{Zn^2+}$ would thus make a poor galvanic cell as most of the reagents would be wasted reacting directly, rather than doing work in the outside circuit.

With a simple porous separator, there are no other ions than the ones in the half-cells, so naturally, these are the ions that flow across, with the goal simply to prevent bulk mixing of the two solutions.