What's a word like "wander" that implies destiny?

Peregrinate
Travel or wander from place to place.

Oxford Dictionary

The reason I suggest it is because it has associations with the concept of peregrinatio in Anglo-Saxon & early Christian literature, 'a very interesting and characteristic form of ascetic discipline known as peregrinatio, 'peregrination,' 'wandering,' literally 'pilgrimage' " (Chadwick 1961, 80). Here it can have connotations of destiny as well (in poems such as The Wanderer).

To see if it fitted with your context, I googled peregrination and Wandering Jew and there is a book, American Folk Legend, which refers to the Wandering Jew's 'peregrinations', see page 24.

You will not find a single word that anwers the question because wander is too far removed from the other meaning you are looking for (purposeful or destined). So you will need to choose a word other than wander to represent the act of walking along the river bank.

verb: wander; 3rd person present: wanders; past tense: wandered; past participle: wandered; gerund or present participle: wandering

1. walk or move in a leisurely or aimless way.
Google Dictionary

Of particular importantance here is the fact that to wander, is to move in a 'leisurely or aimless' way, this is very far removed from a 'definite purpose'... and the idea of 'destiny'.

In the abscence of any similar word to wander that conveys your intended meaning, I came up with the following phrase. This is close to your intended meaning I would say.

"The explorer fervently pursued the contours of the river bank, consumed by wanderlust"

The two key words I included for the purpose of your intended meaning are:

Fervent:

fervent ˈfəːv(ə)nt/ adjective adjective: fervent

1. having or displaying a passionate intensity.
   Google dictionary

and the verb pursued:

to follow someone or something, usually to try to catch him, her, or it:
Cambridge dictionary

the 'usually to try to catch ... it' is key here, as it adds to the sense of definite purpose you are trying to express.

Consider haunt:

Merriam-Webster:

• to visit often
• to have a disquieting or harmful effect on :  trouble
  <problems we ignore now will come back to haunt us>

Macmillan Dictionary:

• to go to a place very often, especially when you are not welcome
• if a place is haunted by the spirit of a dead person, some people believe that it appears there
  They say the prison is haunted by the ghosts of the men who died there.

Collins Dictionary:

• to visit (a place) frequently
• to visit (a person or place) in the form of a ghost

While the definitions refer to going to a place, “the explorer haunted the river” would probably be interpreted to mean that he visited various points along the course of the river; i.e., wandered.  But it takes a direct object, so it should not be used with a preposition like “along”.

While haunt can be used in a perfectly mundane sense, e.g., He haunted the galleries and bars that the artists went to, it also has a supernatural sense, so saying that a (living) person “haunted” a place might be interpreted to mean that he was condemned to do so (although it would help if you reinforced this with some context).

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."

OR

"The explorer was impelled along the river, consumed with wanderlust."

Explanation

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:

1. 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)."

2. Your sample sentence uses "wanderlust", which has nothing to do with destiny. It's a passion rather than a fate.

3. 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.

Inexorably
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.

Impel
Drive, force, or urge (someone) to do something:
"financial difficulties impelled him to desperate measures"

Some other verbs to consider: propelled, compelled, driven.