In English, and some other Germanic languages, prepositions have become a very article-like thing that in some positions appear before infinitives. The distribution often differs from that of articles, but the idea is quite similar. (Both have interesting quirks in English - Separately from English, this has also developed in other Germanic dialects, even far enough to be separated from the rest of the 'to'-area by other constructions (e.g. despite Swedish having a non-cognate 'att' for the same role, some north Swedish dialects too use cognates to 'to'; 'att', however, also originates with a preposition.)
Now, several families subfamilies in the Indo-European clade have a feature whereby verb roots combine with prefixes that are quite clearly prepositions in the language. These combinations may form even rather opaque meanings:
The way the preposition affects the meaning of the verb is not really obvious in any of these examples. Now, an interesting development of this is how it's interacted with aspect in the Slavic family of languages.
However, we can go on and consider a situation whereby prepositions do not combine with verb stems and thus forming new lexemes (as in the IE examples). What if, instead of prepositions/adverbs* merging with verbs to form lexemes, we had prepositions merging with infinitives to form some TAMs (and also the potentially tense-, aspect- and moodless infinitive). It's easy imagining a preposition marking an imperative ('for', anyone?), another marking progressive tense ('at', perhaps?) and one marking the basic infinitive.
Another thing one could do is have the infinitive marker be lexically specified by the verb; if they have a similar origin as in English, one could imagine something like
to eatpossibly distinguished by type of action (cognition vs. kinetic vs. passive vs. ...) or by some lexical feature (inherent aspect), or even permitting some distinction to be made by choice of preposition.
This would, anyway, make for an interesting similar-but-different development as to what has happened in several Indo-European branches.
* IE prepositions originated as adverbs that apparently could modify verbs as well as nouns in oblique cases, and only later got more closely bound to the nouns. In many IE languages, they can still be used "intransitively", and as adverbs - English being a trivial example of this.