TL;DR: there are two broad options, neither of which exactly fit your sample sentence but both giving the required sense:
"The explorer wandered inexorably along the river."
"The explorer was impelled along the river, consumed with
There are some issues in the way this question is framed that make it difficult to answer, but by working through these, a suitable solution might be at hand:
As Gary has commented, "A problem you have here, is the word 'wander' actually is contrary to the other meaning you are looking for (definite purpose, or destiny)."
Your sample sentence uses "wanderlust", which has nothing to do with destiny. It's a passion rather than a fate.
Your sample sentence is in the active voice ("the explorer xxx-ed") whereas both your explanation and the connotation of external forces acting on the explorer really urge the passive voice for your verb: "the explorer was (condemned / doomed / blessed) to wander".
If wanderlust is dispensable in your sentence but you want to maintain the active voice, then using an adverb with wandered to provide the sense of destiny might be the best solution. Perhaps inexorably will do the trick.
In a way that is impossible to stop or prevent:
"the conflict was to lead inexorably to the outbreak of World War I"
Some other adverbs to consider (my preferences in italics): inescapably, inevitably, obdurately, implacably, unceasingly, unendingly, unremittingly, unrelentingly.
If wanderlust is important but the voice of the verb is not, then a verb in the passive voice and suggesting an external agent or irresistable force might be preferred - for example, was impelled.
Drive, force, or urge (someone) to do something:
"financial difficulties impelled him to desperate measures"
Some other verbs to consider: propelled, compelled, driven.