When can a deterministic finite-state-automaton (DFSA) along with its input sequence be said to be a part of another DFSA?
It's not clear to me how to interpret &quot;can be considered&quot;, so I'm going to identify one technical question that can be answered.
Given a FSM $F$ and an input sequence $S$, it is possible to build another FSM $F'$ so that the execution of $F'$ on the empty input is in one-to-one correspondence with the execution of $F$ on $S$ (and in particular, both end at the same end state(s); e.g., either $F$ accepts on $S$ and $F'$ accepts on the empty input; or $F$ rejects on $S$ and $F'$ rejects on the empty input).
The proof is a straightforward application of the product construction: we construct one FSM $F_0$ that outputs the fixed sequence $S$, and then compute the parallel composition of $F_0$ with $F$.
The following is also true: given a Turing machine $T$ and a fixed input $P$ (i.e., initial state of the tape $P$), then it is possible to construct another Turing machine $T'$ such that execution of $T$ on input $P$ has the same result as execution of $T'$ on any input.
Formal proofs with Turing machines are often tedious and uninformative, so it's easier to see how this is true by considering a real-world program.  For instance, suppose we have Python code that defines some function t():
def t(x):
    ...

and we have some fixed string p.  Consider the following Python function t':
def t(x):
    ...
def t'(x):
    return t(p)

Then it is easy to see that the behavior of t' on any input is equivalent to the behavior of t on input p.  (Here we have hard-coded a lexical constant string in the place indicated with p above.)  You can do the same thing with Turing machines, where the machine first writes $P$ on the tape, and then starts executing $T$, to define a Turing machine $T'$ that proves the claim above.