Dolphins won two Super Bowls in the early 70s, and the 90s were good (when they had Dan Marino at QB) and late 70s/early 80s they were decent also.
The Dolphins are a weird case. Since they're in Miami they have a bit of a "fair-weather" fan-base. Not saying you are that way of course (obviously, you can't be since they've only been to playoffs twice since 2002).
Their PL equivalent would probably have to match the below criteria (in my opinion):
"Weak" fan-base and located in a vibrant city/town
Some success in history but recently unsuccessful
Usually only make it pass the first round in the cup, but they have been to few cup finals.
Usually have one standout player that makes things happen.
Probably a lower table side but usually avoid relegation as there are other teams worse than them.