Are their orbits circular as Neptune's (in this case) or highly
L4 and L5 stability is rather narrow
A more elliptical or eccentric orbit in resonance with a planet with a circular orbit would reach well outside of L4 and L5 and wouldn't be an L4 or L5 object. (though I have to admit, a planet's eccentricity and the effect that has on L4 and L5 stable zones has gotten me curious), but they would likely need to be pretty close to the same eccentricity or it wouldn't be L4 or L5.
Earth's curious orbital partner 3753 Cruithne probobly isn't stable, long term anyway. It's elongated as you describe and it could be stable for a long time, perhaps millions of years but it's not an L4 or L5. 3753 Cruithne has an orbit around the sun of 364 days, so it, in effect, laps the earth every 364 years, rarely coming close enough to hit or significantly change it's orbit cause it's on a different orbital plane.
For an object go from L4, past L3 into L5 it would need to be orbiting slightly faster than the planet.
Now, that object oscillates between L4 and L5, that would be more exotic and much harder to explain. But passing though L3 is possible if the object is orbiting slightly slower or slightly faster than the orbit of the planet. It could be a Jupiter or other planet's resonance that's causing it, but that's just a guess.
How could an object in the same orbit as Neptune have another orbital
period than Neptune? Or how else can it migrate between L3, L4 and L5?
Saying an L4 or L5 object has exactly the same orbit as the planet is probobly inaccurate. The planet tugs on the L4 or L5 object anyway, so there's kind of a dance with Trojan objects. It might average out to being the same orbit, but it's never precisely the same on any given pass. Tugs from other planets affect the dance too. Jupiter, Saturn and Uranus all probobly tug on Neptune's Trojans, throwing off the clockwork a bit.
And are "trojan" objects expected to gather in L3 too?
Gather, no. Pass through, yes. L3 isn't stable, but that doesn't mean an orbit couldn't pass through L3 and be in a near L3 Orbit for several rotations around the sun, but it's not stable. An L3 orbit would tend to move away from L3. L1, L2 and L3 aren't stable, but they are still good points to put a satellite because less energy is needed to keep a satelite in those regions.
Hope that's not too clumsy an answer. My layman's attempt. I can try to clean up if needed.
Edit / Update
2010 EN65, per your original link is a co-orbital with Neptune, so it's not a 3/2 or 2/3 or any other fraction it's 1 orbit to 1 orbit.
It's also a Centaur, which means it's long term orbit is not stable. A true Trojan asteroid is stable.
I found an article on Jumping Mechanisms of Trojan Asteroids here - it gives a brief summary but it's a pay article: http://link.springer.com/article/10.1007%2Fs10569-015-9609-4
and a map of EN65's orbit (has to be calculated, it's not been observed for nearly that long), but map below:
It's pattern looks similar to some of Earth's co-orbital asteroids, including 3753 Cruithne that I mentioned above.
diagrams here: http://18.104.22.168/petercaspari/bdi/coorb.htm
So, that's probobly a more accurate comparison than Hildas. Look at 3753 Cruithne, or other planet's co-orbitals.
What confuses me - and I hope you forgive me for only enough to be dangerous, as they say, but what confuses me is that your original link says it's an elliptical orbit but it also seems to be drawn to both L4 and L5, spending more time there, so I might have been incorrect in suggesting the elliptical orbit wouldn't be possible. - sorry bout that. At least, some additional eccentricity seems likely, but I'm guessing, probobly not a huge amount. If I was to guess, I'd guess that EN65 is less eccentric compared to Neptune than 3753 Cruithne is compared to Earth.
Here's a cool picture of Jupiter's Trojans (Jupiter's the King of Trojans, it's got like 50,000 in both L4 and L5)
and you'll probobly notice, there's one point at (or, perhaps more accurately, passing through) L3.