As you've read, different schools have different expectations. My son in third grade was doing spelling words like "overlook" which, while it's a big word, it is made up of two easier words. My friend's son in 2nd grade in another state was doing words like "abbreviation"!!! He has more homework than my son has in 3rd grade too.
College can be the same way. Back home a friend was going to a very well known university for nursing. 2 years in she switched to the private college in the country I was going to. Guess what? Almost a year's worth of her classes didn't count because they tested her on entrance and she didn't score high enough.
The schools CAN base things on an entrance test. I had to take one for high school AND college so while transcripts matter some, the entrance test matters more because that is based on what the current school expects.
I would find out what he is considered behind on and see if they would be willing to retest him at the end of the summer. Then you can spend the summer working with him to bring him up to speed. If you can do so, great and they should move him to 2nd grade. If not, he needs to repeat 1st because if he doesn't have a strong foundation, it is going to affect everything else he learns from here on out. He MUST have a strong foundation or like a house, he will crumble as school gets harder.
Whatever you do, please don't make a big deal about this in front of him. Be careful what you say so as not to make it seem like it is his fault or he isn't smart enough or whatever. The reality is, your son is just as smart today as he was 2 weeks ago. He probably just hasn't learned some of the stuff yet that they tested him on because of the difference between schools. It's very important you reassure him that everything is okay no matter how things turn out!!