What you're getting at is clear, but it's coming from a fundamental misconception of entanglement and, more generally, the concept of superposition. This is a subtle concept, well outside our normal experience, so don't feel bad!
Without using any math, it's a bit difficult to explain, but here's the general idea: The particles exist in a correlated superposition of states. The superposition bit means that each particle can be (for the sake of example) spin up or spin down when measured. Before being measured, the particle is, in the "language" of quantum mechanics, in both the spin up and spin down state simultaneously. It's not switching/cycling/jumping between the states in any way. Remember that for a statement to have meaning, it has to tie in to a measurable experimental outcome. Before you measure, we can only talk about possible outcomes, and the way most people think of this situation of could-be-one-or-the-other is that the particle is in both states up until it's measured. This is more a philosophy than anything, since it matches the math we use, but it can't be measured before it's measured, if you follow my drift. So you can't talk about the particle cycling between two states because you can't measure any such cycling. (One clarifying note: though the idea that it's in both states simultaneously isn't measurable, the fact that it's not in one state or the other state is something we can measure indirectly. That's why we tend to say it's in the up and down state - because "or" would be wrong. Language has a hard time keeping up with quantum.)
Now, in a quantum entangled state, the only difference is that the superposition of states of the two particles is correlated. Correlation means that if you measure one to be spin up, the other will be measured as spin down. Correlation does not imply causation, i.e., measuring one particle as spin up does not cause the other particle to choose spin down, any more than pulling a right-handed glove out of a pocket causes the other glove in the pocket to be left-handed. The two are correlated, but there's no causation involved.
So, time doesn't come into the equation of entangled particles in the sense that you're asking. If they remain entangled (not an easy state to maintain, by the way) and are in very different physical situations (say, one in a gravity field and one not), they still remain correlated. There is no cycle to become out of sync, which pretty much is the end of the story.
I do want to mention that this answer may feel very unsatisfying. Someone may give you a more mathematical treatment of the answer, though I doubt that'll help the nagging feeling that it just can't be that way or that it doesn't make sense (even though you can follow the logic). Unfortunately, this is pretty much the standard outcome with any of quantum mechanic's more esoteric properties. At some point, you just have to accept that the way the world works at small scales is fundamentally different than the expectations we form from our experience at human-size scales.