I was wondering if there are there situations in research wherein equivalence between the treatment and control groups is dependent on time... for instance could there be situations wherein the equivalence between the treatment and control groups may disappear with time after the randomization process?
One famous example that comes to mind is Ashenfelter's Dip, named the US economist and oenophile Orley Ashenfelter. Imagine that you have a job training program with good randomization. Ashenfelter noted that the mean earnings in the treatment group often dip before the start of the program, leading to imbalance with the control group.
Other common problem are attrition from the experiment that depends on the arm, or when some control group members seek out another program similar to treatment (substitution bias).
If you want to make an intent to treat analysis, then the groups are always comparable.
But almost always that's not what you literally want. You usually want to analyze the treatment itself. Then, the answer is that basically all randomization schemes will suffer from non-comparability of treated and control in some degree, due to non-compliance, loss to follow-up and basically any usual problems that can happen in observational research, because you are not under strict control of the treatment assignment anymore. It's usually expected that the longer the study takes, the harder it will be to keep enforcing protocol during time, all else equal.