Some Indo-European languages have a construction called go-periphrasis, by which some form of the verb go is used in conjunction with the main verb to mark tense. Most languages that have this feature use it to mark the future tense, like English (going to), Portuguese, Spanish, Dutch and French. Catalan uses it to mark the perfective past. There are probably others, but I could not find any example of the go-periphrasis outside the Indo-European family.
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The use of motion verbs to mark tense and/or aspect and/or modality is very common across the world's languages, both as part of serial verb constructions or as auxiliary verbs.
Below are some non-Indo-European language examples.
Sùpyìré, Niger-Congo family. Serial verb construction with the verb sí 'go' indicating futurity:
(1) Zànhe sí dùfugé kɛ̀ ɛ̀ gɛ̀ rain go maize.DEF spoil "The rain will spoil the maize."
Bambara, Niger-Congo family. Here the motion verb na 'come' is used as an auxiliary indicating futurity:
(2) a na taa 3sg come go "He/she will go."
Grammaticalisation of the motion verbs 'go' and 'come' as future markers is common in Bantu languages. More examples in this paper.
Squliq Atayal, Austronesian family, Taiwan. Here we see musa' 'go' used as an auxiliary to indicate irrealis/future:
(3) musa’ m-nbu’ yaya’ =mu IRR AF-ill mother =1S.GEN "My mother will be ill"
George Lakoff (1993: 218) suggests that the metaphorical basis of describing time in terms of space is biologically determined: “In our visual systems, we have detectors for motion and detectors for objects/locations. We do not have detectors for time (whatever that could mean). Thus, it makes good biological sense that time should be understood in terms of things and motion.”
I'm not sure if this deserves to be an answer in its own right or if it should just be a comment, because I don't think it's exactly what you're looking for. It involves aspect marking instead of tense marking...
Japanese uses the constructions -te-iku and -te-kuru as aspect markers on verbs. When used as verbs in their own right, iku and kuru express 'go' and 'come', respectively. When used in combination with other verbs (with a verb "linker" -te-), they can either retain their original meanings in tandem with the other verb or serve a purely grammatical function. In the latter case, -te-iku expresses a kind of continuation of an action or a state from now on, while -te-kuru expresses actions or states that have been continuous until now. The following examples were taken from this online Japanese tutorial, one of many informative lessons on a website called "Yasuko's Nihongo House"! (Note that kita is the past tense of kuru.)
I have seen these markers glossed simply as -IKU and -KURU in syntax and semantics examples, but Matsuo Soga calls this aspect ingressive aspect in his book Tense and aspect in modern colloquial Japanese.