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In English (for example), we say "I go/went/was going/etc.", inflecting the verb for tense and aspect while leaving the subject of the sentence unchanged. But are there any languages that would instead inflect the noun/pronoun? For example, suppose English were to have a nominal inflectional mark "-tu" indicating past tense: "Shetu go" instead of "she went."

Or are there at least languages where the noun must agree with the verb and not the other way around? So, using the made-up inflectional marker from earlier, we might say "Shetu went" instead of either "Shetu go" or "She went."

And if not, why not? It seems particularly strange (or "asymmetric") given that (in many languages), nominal inflectional categories like person or number affect the inflection of the verb (via concord).

There is a fair amount of literature referencing languages that mark tense on nouns (e.g. this Tonhauser paper: https://www.asc.ohio-state.edu/tonhauser.1/papers/sub9tonhauser.pdf). From what I remember, the meaning of a past-tense-marked noun may be along the lines of “an ex-/former [noun]”; I don’t think it’s common for clausal tense to be marked on a non-pronominal noun in the clause.

There are also languages that use different “subject pronouns” for different tenses of the associated clause.

Here is a relevant Wikipedia article: https://en.m.wikipedia.org/wiki/Nominal_TAM