I believe it is more of the fact that the organism gets stuck to the residual resin on the plant itself, and becomes encased and later fossilised in the amber.
A host of bugs, fungus and other life forms have been found trapped in amber from the time of the dinosaurs. Bugs can become encased in amber if, while alive, they get stuck in the resin.
Although plant resin is slow-moving, the bug stands to be encased in it if it is already stuck to the tree itself, as there is a large amount of resin on the surface of the plant. If they are crawling over the plant, or land on the surface, they would become stuck and later encased. It would be similar with other organisms as well, as plants and other organisms have been discovered preserved in this way.
These fossils represent some of the earliest African fossil records for a variety of arthropods, including wasps, barklice, moths, beetles, a primitive ant, a rare insect called a zorapteran, and a sheet-web weaving spider. Parasitic fungi that lived on the resin-bearing trees were also found, as well as filaments of bacteria and the remains of flowering plants and ferns.