Coat sometimes greasy, sometimes dry, sometimes nice & shiney

One of my cats (Onyx) (who is living with my sister at present) seems to go thru cycles with his coat, it will be greasy for a few days, then be nice and shiny  for a few days, then it might be dry (no dandruff) for a couple of days, then greasy or nice & shiny again....  My sisters cat seems to go thru the same cycle... They are both DSH cats, aged 13 & 14. My boy has arthritis also...

I was looking into what might be making his coat this way and seemed to only be able to find cats who's coat is dry, or greasy not a cycle of all 3...

Initially I thought it could be his arthritis making it painful to groom, but as it's affecting both cats I'm thinking it could be a diet issue.... His diet could be better he's just on supermarket food (Chef - an NZ made brand, with Optimum biscuits)... I want to change his diet regardless of it being the cause or not.... Since having health issues with my other boy Zap (HCM) I've read a lot about cat food and feel awful for nearly everything I ever fed them! So I would prefer him on a good cat food, he seems to really like Wellness Core, and also Primal (the NZ Primal canned food - good stuff, different from the Primal frozen/freeze dried sold in the USA), he will also eat Science Diet biscuits tho I'm not sure who or what to believe in regards to the Hills products...

His regular check up at the vet not long ago didn't reveal any new issues, and this isn't something that has just happened. I talked to my sister and said I thought it would be a good idea for me to take my boy to the vet for a checkup and she told me that they both have this "issue" and that she's not worried about their health as the check up went ok... I'm still not quite comfortable with that... Can anyone offer any advice or suggestions?? My understanding is a cat should generally always have a nice shiny healthy looking & feeling coat.... this business just doesn't seem right to me, am I over reacting??

I don't think you are over reacting.   A cat's coat can tell a story.  It should be clean and shiny and if it's not, there's a reason.  A cat with arthritis may not be able to groom ALL areas of his body, so their may be just certain parts that look a little ragged, but overall he should still look fairly well kept.  BTW, do you give him anything for the arthritis?  If it's not too advanced, you can give him Cosequin or Dasequin mixed into his (wet) food.  You can get that via Amazon.  Or krill oil or green lipped mussel oil or powder is supposed to be good too.  OR ground up chicken feet (that's supposed to be REALLY good, and it's all natural, since you want to feed him better, 'course, that would actually be his food...just an additive)  

Adding some type of Omega 3 Oil to his food would be great for his skin and coat anyway, and would probably help with his arthritis as well, depending on which oil. 

OH!  I just looked up the Primal NZ foods, and see they contain Omega Oils AND Green Lipped Mussel.  That sounds MUCH better than a grocery store brand would be