You could always do a "cinematic universe" in a more unique way, rather than focus on multiples ships and crews at one time, why not focus on one ship through it's service. That way you could have several crews carrying on the ship's story as it goes through a time of great change, even with an underlying plot where each crew deals with elements of what's going on.
For example, the Miranda-Class U.S.S. Diligent is commissioned in 2274 and heads out on her maiden voyage, a long-range survey mission. Jump forward a decade when the Klingons are rattling sabres in response to Project Genesis and a standoff along the border. Then to the signing of the Khitomer Accords when they find themselves assigned to a good will mission to support the Klingon Homeworld following the destruction of Praxis. Then into the early 2300s, dealing with tensions between the three major powers. Then to 2311 and the Tomed Incident. The 2330s and rising tensions with the Cardassians and Tzenkethi. Ending in 2344 with the ship about to be decommissioned after 70 years of service, though not before being the second ship to reach Narendra III after the battle claimed the E-C. You could then have a second Diligent being commissioned soon after, a newer vessel such as a Cheyenne- or Springfield-Class, taking up the mantle and carrying on the story over the next 30-40 years.
