Connected Speech Processes
The OP's question is dealt with by what is called connected speech processes. Below is list of the most common features for final /t/ in English. 4) is the one being asked about here.
These are from a list for the /t/ at the end of a word, they are said to be assimilated [basically, become completely attenuated]
1) *Word-final /t, d, n/ become bilabial before bilabial consonants.
[my example: dirtbag, street boy]
2) *Word-final /t, d, n/ become velar before velar plosives.
[cat gut or bright cat] [velar plosive here: the /g/ as in gut, and the /k/ as in cat
3) *Word-final /t, d, s, z/ become post-alveolar before the post-alveolar phonemic approximant /j/, with possible disappearance of /j/.
[[also written as ⟨ɹ̠⟩ in IPA, and ⟨ɹ⟩, ease of typing. [this is the r in red by the way]] [[the transcription of this post-alveolar approximant is discussed at length on Wikipedia, but not worth going into here.]
[ light room or street route]
Also for the /t/: there is elision [omitted]:
4) Word-final /t, d/ preceded by a consonant and followed by a consonant in word-initial position are omitted.
[lift the cat or trust the process]
[full list ] from: Table 2 summarizes all the processes affecting pronunciation in rapidly articulated connected speech that we have discussed in this section.1
I thought it might help the OP to have a simplified list. Also, here is a longer article on the subject along with what is called co-articulation, which is explained in some detail in the article: connected speech and co-articulation
If my examples contain errors, I'd be glad to fix them. Though I am a linguist, I am not a phonetician and, therefore, am probably prone to slips between my ear and my writing.