"The smart bear is running away from those bad people and hurrying into the cave."

Translation: Az okos medve elfut azoktól a rossz emberektől és besiet a barlangba.

[JanTatouse]

Is it absolutely necessary to use besiet in this situation (as opposed to siet)?

[bdori]

I think it's better with besiet. "Besiet" means that the bear runs into the cave, "siet" would mean that it runs in the direction of the cave (and maybe goes inside).