If I walk up to a pier and throw kool aid into the ocean, where will those kool aid particles end up on a long enough time scale? Is it arbitrary? Will they travel the entire ocean at one point?
It depends on what is thrown in, how much of it is thrown in where it is thrown in and the ocean currents in the region and the chemical properties and reactivity of what has been thrown into the ocean.
To paraphrase researchers from Woods Hole Oceanographic Institute, anything that is dissoluble in sea water begins mixing with sea water and becomes increasing dilute. Depending on how soluble the substance is, it may react with either the water or with other molecules in the water to form alternate compounds. As the substance moves offshore it will then be taken across the ocean by currents. Depending on the longevity and traceability of the compounds they could end in most of the oceans of the world.
If you threw a small amount of cool aid in the ocean it would quickly disperse. A larger amount would take longer to disperse.
Radioactive elements from the Fukushima power plant that exploded after being damaged by a tsunami in 2011 took 2.1 years to reach the western shores of North American.