To start, if I'm reading your question correctly you've got the general galaxy evolution model backwards, at least as far as morphology goes. At a very high level the picture goes like this: most if not all (large) galaxies form as spirals, then at varying points in their lives they merge with other large galaxies (either ellipticals or other spirals) and the product becomes an elliptical galaxy. This happens faster and more often in dense environments like galaxy clusters, and takes longer to occur for more isolated galaxies like the Milky Way (which will merge with Andromeda in a few billion years).
Question on isolated spirals
If what you were asking about is isolated spirals, well no I don't think an isolated spiral would ever turn into an elliptical galaxy by itself (at least not on life-age-of-the-Universe timescales). It may eventually start to resemble an elliptical galaxy (in that its stellar halo may grow) depending on how much it gets harassed by other small dwarf galaxies. No galaxy is truly isolated, there is always a hierarchy of smaller galaxies nearby and they gravitationally interact with the host galaxy. For example the Milky Way has the Sagittarius dwarf galaxy, and the Large and Small Magellanic clouds which will merge with us soon (it's basically happening now for Sagittarius, and the Magellanic clouds will be a few billion years). None of these events will be impactful enough to turn the Milky Way into a true elliptical, that will only occur with it merges with Andromeda.
Question on morphology versus internal dynamics
To specifically answer your other question on how morphology relates to the internal dynamics of a spiral galaxy: it most definitely does, but the relationship is complicated. When you see a spiral galaxy with a central bar, or spiral arms, these features are being generated by dynamical instabilities within the galaxy. This could happen thanks to a number of causes, which may operate from within the galaxy itself, or from outside the galaxy. One of the most common means of exciting a bar or spiral arms in a disk galaxy is by interactions with smaller dwarf satellite galaxies that orbit around the larger spiral galaxy and gravitationally perturb it. If this part of the answer sounds like a cop out, well it sort of is. Exactly how disk perturbations like bars and spiral arms are started, how they work, what the heck they do to the galaxy, these are all truly bleeding edge research topics that astronomers are working on right now.
Hope that helps!