Like I said previously, early growth soft leafy grass, would be better to slowly introduce into the diet and balance out with a more mature fibrous grass hay.
When I talk about fresh grass, this isn't just lawn grass that I'm talking about, but thicker leaf grasses like bermuda, orchard, meadow, etc, that you will see growing in the wild, or as weeds, or in crops. And this thicker leaf weed type fresh grass that is already at a more mature growth stage where the coarse stem is starting to grow or has grown to start a seed head, is just like the hay you buy, but just not dried. Therefore doesn't need to be gradually introduced into the diet.
Fresh grass isn't just lawn grass like kentucky bluegrass. If you're thinking of grass as being lawn grass like kentucky bluegrass, then it's almost always going to be a rich soft growth, so yes, does need to be introduced gradually and limited, balanced with a coarser hay.