When English voiceless stops are aspirated, a period of turbulent noise occurs after the release of the stop. This turbulence does not come in between the release of the stop and the articulation of the segment that "follows" the stop (the segment that comes to the right of the stop in a phonemic transcription). Instead, it overlaps with (or modifies) the "following" sound. When a vowel follows, the period of overlap can be (but usually isn't) transcribed phonetically as a devoiced vowel, e.g. pin = [pʰɪn] = [pɪ̥͡ɪn]. I'm writing ɪ̥͡ɪ here to represent that the overlap is only partial; the voiceless sound transitions into a voiced vowel sound. Similarly, /h/ by itself could alternatively be transcribed as a devoiced version of the following vowel; e.g. hiss = [hɪs] = [ɪ̥͡ɪs].
When a glide or a liquid follows a syllable-initial voiceless plosive, the period of aspiration overlaps with the glide or liquid, so a word like "twin" can be transcribed as [tʰwɪn] or alternatively as [tw̥ɪn]. Even more specifically, I believe some phoneticians consider all audible devoiced vowels, glides and liquids to be phonetic fricatives, and in the case of [w], there is a special symbol in IPA for its voiceless fricative counterpart: [ʍ]. So probably it could also be transcribed as [tʍɪn], or
even as [tʷʍ͡wɪn] if you wanted to indicate very explictly a pronunciation where the rounding extends before and after the period of voiceless frication. Likewise, [j̥] (voiceless [j]) is more or less the same thing as the palatal fricative [ç], so words like hue and pew can be transcribed as [çu] and [pçu] (I'm using [u] as a broad representation of the vowel, since we aren't interested in its exact quality).
There is a linguistics.se question about the topic of whether "voiceless approximant" = "fricative", which is apparently debated: Do voiceless approximants exist? What is the consensus among phoneticians/phonologists?
So a word like print can be transcribed as [pʰɹɪnt] = [pɹ̥ɪnt] (or possibly [pɹ̥͡ɹɪnt], if the /r/ is only partially devoiced). I'm using [ɹ] broadly here because I don't know enough about the exact articulation of English /r/ to give a narrow transcription. I don't know of a special symbol like [ʍ] or [ç] for the voiceless fricative counterpart of [ɹ].
A relevant Language Log post:
"Hwæt about WH?" April 13, 2011, by Mark Liberman. I think the following quote, although it is not about aspirated stops, may answer your question:
In utterance-initial American English /h/, there's generally a short but well-defined voiceless period, and then the larynx is adjusted so that the turbulent flow is replaced by regular oscillation (i.e. "voicing") in the body of the vowel. But this laryngeal maneuver doesn't constrain the rest of the vocal tract — the lips, tongue, velum etc. are free to do whatever.
Since aspiration is about what the larynx is doing, the tongue is free to move for the /r/ at the same time or even before the aspiration starts. The tongue does not wait until after a period of aspiration is finished before moving for the /r/.