Switching from a newer version of Twine to an older one can be done, though you may experience odd behavior in the transition. Old versions of Twine of course aren't aware of any new capabilities that the newer version has added.
Twine follows semantic versioning. This means that the first version number of Twine only changes when a backwards-incompatible change is made, either to how it saves stories or how it interacts with story format extensions. This means that going from (using hypothetical versions for the sake of argument) Twine 15.0.0 to Twine 14.3.6 will likely be tricky, but going from Twine 14.3.6 to Twine 14.2.0 will not be.
What exactly you experience will depend on the particulars of the versions you are moving between. You may lose data in the transition. Switching from a newer version of Twine to an older one, generally speaking, is not well-tested.
Regardless of whether you are using browser Twine or app Twine, you should save an archive of your work. This will ensure that even if the absolute worst happens, you have a safe copy of your work.
Each version of Twine since 2.0 is available by visiting
https://twinery.org/[version number] in your browser. For example, you can use
Twine 2.1.1 by going to
Older releases are available on the Twine GitHub repository.
If you are using a built-in story format (Chapbook, Harlowe, Snowman, or SugarCube), then you will almost certainly need to either install a newer version of the story format separately, or change the story format your work uses.
You may also need to reset your preferences. Although the instructions linked are for the most recent version of Twine, they will probably work with most older versions of Twine as well.