Aaron Bergman had something like this in his "World of Endings Promised." The journey across dimensions granted the Wanderers immortality as well as their abilities... until something went wrong.
My take is that Earth, El-Hazard and Creteria are the same planet in three alternate/what-if dimensions. All three worlds have humans, and although many species are different, they all have basically-compatible animal and plant life.
Only a (cough) complete weirdo would write a semi-realistic story with a El-Hazard native who contracts terminal cancer. (BTW, I just wrote about 1000 words of Hana Ni Arashi. First real progress on that darn thing in ten months. Woo hoo!)
Another author, Alan Harnum, wrote a piece (still unfinished) titled "Mortal Engines" which addressed this issue... but from a different angle.
(yeah.. I know... you have it up on your website... heh)
Personally I ascribe to his view of what happens to Earthlings as they stay longer and longer on ElHazard... that the powers they have inhierited gradually alter them as time progresses.
regardless... as to diseases.. I'd think that either earthlings would be very susceptable to Elhazardian diseases(if they are human just like us)... and vice versa... or... they'd be completely immune (and vice versa) because of significant physiological/genetic structures. IE...just as you can't catch feline leukemia from your cat, Elhazardians may not actually be "human" at a genetic level.. they may simply look human and seem to have identical physiology as humans... but use entirely different DNA and RNA strings... and hence cannot catch human diseases at all.
Of course El Hazard could be a colony of a space faring civilization ( note the dual suns... very unlikely that a terran standard wolrd would be able to exist in a solar system with a binary star system.. mostly because the widely fluctuating solar energy would create wildly oscilating solar winds, creating hotter and colder cycles on the planet. Terraforming technology like nano tech would be needed to moderate such effects. If this is so, it's likely the elhazardians would have medical nanotech that would make them supremely healthy (they would never catch diseases), and it could be that such nanotech would exist in the food they eat and water they drink ... which would have also been introduced into the Earthlings as well since they've obviously eaten and drank
Anyway.. that said.. I don't think there was any indication of sickness on Elhazard at any point in any of the OVA's or TV shows.
Anyway... back to the point I was gonna make.
The powers of the earthlings seem to be growing as they stay on ElHazard.. at least to me they do... and if so...it's likely that they are changing for a reason.
Specifically, if you note, even when Fujisawa is pretty 'dry' at points early in the story... he never gets as strong as he did eventually when fighting the super bugrom. This seems to indicate that Fuijisawa's ability became stronger over time.
Likewise, Mokoto's interfacing ability seemed to get stronger as well. Specifically, he went from just grazing through Ifurita's memory store initially, to being able to communicate with her in her memories... to finally being able to delve into her core systems and destroy her obedience circuit.
That said.. a case could be made for this also being simply that they became more adept at using their abilities... but that doesn't seem to mesh well with Mr. Fujisawa's purely physical abilities. Seems to me he'd have had his glowy strength field before the time he exhibited it if this were the case.
Anyway... Alan Harnum seemed to be saying (in Mortal Engines) that all the earthlings were gradually becoming something more than human... which I think I agree with.
in that vein I reccomend taking a good hard look at the episode where Ifurita is awakened by Jinai, and fires a shot a Makoto... hiting the ground right in front of him. Take a good hard look at the explosion and the after affect.
You'll see that the explosion seems to wash along a spherical "safe" area around Makoto, looking rather like the force bubble that demongods are supposed to have around them according to the manga.