Some of the shifts are obvious such as palatalization, or any consonants with adjacent places and manners, but some other more radical changes are also observed in unrelated languages. For example, Spanish has the vowel breaking of o into ue as well as changing a typically palato-aveolar j into a velar j /x/. Compare that to the Mandarin example of 约 which is pronounced /jyɛ/ in Beijing Mandarin and /jɔ:/ in Sichuanese Mandarin, and 鞋 which is /ʃiɛ/ in Beijing Mandarin and /xai/ in Sichuanese Mandarin (which also resembles the Great Vowel Shift). Given that the chronology of Chinese pronunciation is quite a bit murkier than the Romance languages, maybe "shift" is not the right word to use, but these examples do exhibit a very similar pattern of phonological differences among related languages even with very radical changes. Is there any specific hypothesis as to how pronunciation is passed down and around that would give rise to these patterns across unrelated languages?
I assume you aren't just interested in the specific cases of Chinese and Mandarin. The primary reason for any sound shifts is that at some point in history, the actual pronunciation of a given sound becomes ambiguous, to the point that children learning the language can't really tell which sound was uttered. For example, in most dialects of American English, /t/ at the end of a syllable is produced "unreleased", so the only acoustic clue that "heat" ends with a /t/ is a pattern of formant transition. If you release the /t/, it is much clearer from the burst of /t/ that there is a final /t/; without the burst, it's hard to tell final /t/ apart from glottal stop. And some people actually pronounce final /t/ as a glottal stop -- they have started a historical shift in syllable-final position from t to ʔ.
Likewise, one of the most common sound shifts is from k to tʃ before from vowels. It's almost universal that the tongue has to advance a bit when making /k/ before a front vowel, and often we don't notice that little bit of fronting. But sometimes people do, and make that fronting a bit more exaggerated, which results in a palatal stop (IPA [c]). That [c] sounds a lot more like [tʃ] than it sounds like [k], even though [c] is articulatorily more like [k] than it is like [tʃ].
Mark Hale in Historical Linguistics: Theory and Method gives a good account of how sound changes originate from perceptual and analytical ambiguities in speech, which leads to sound changes. The reason why there are considerable similarities in sound changes across languages is that the articulatory and perceptual seeds of sound changes are consequences of anatomy, not grammar: grammar enters the picture when children across generations make different categorial decisions about how to interpret the facts of speech.
The shifts are due to phonological processes which are universal to humans. So the answer to your question is that these unrelated languages are all learned and spoken by humans -- that is the common denominator.
There are two phonological theories I know of that say this, more or less: David Stampe's theory of Natural Phonology (which I agree with, mostly), and the markedness theory proposed by Chomsky and Halle in chapter nine of The Sound Pattern of English (which is interesting, but wrong). In the SPE theory, the universal part of human phonological systems is the set of markedness conventions, which interpret what happens when the pronunciations of a language lose their special traditional character, and the articulatory system is left to make the best of it, so to speak.
In Natural Phonology, one important source of historical sound change is imperfect acquisition of a language by children. Children must learn not to apply those universal phonetic processes which are inconsistent with the traditional language they are trying to learn. When they slip up, a process which the previous generation managed to learn not to apply, now comes to apply, and we observe a sound change.
Such sound changes are due to phonetic processes universal to human articulatory systems, so we expect the same sound changes to keep cropping up in languages, regardless of their genetic relationship. And that's the way it happens.