Suppose you have 1 future cash flow. You get some fixed constant times the value of the inflation index observed near the time of the cash flow divided by the value of the index observed some time in the past. Most inflation linkers work like this. Some countries have more weird inflation linkers, but we don't need to go there. If you discount this future cash flow using nominal (not inflation-adjusted) interest rates, then the present value of this cash flow is sensitive to these rates too.
In many markets, the the inflation index is known every day, and also there are instruments with observable quotes (like inflation swaps, and inflation-linked bonds) that can tell you what the market thinks the inflation index will be every day in the future. (In some markets, the daily index is publushed for the next few days, so only the index after a few days is still uncertain).
When the value of inflation index on cash flow date changes day to day according to the prediction of the inflation swap curves from prior day, I would call it rolldown, rather than carry. And if it moves it ways other than predicted, then you have P&L because you're long inflation index on cash flow date.
In some markets with a history of deflation, inflation products often come with some embedded floor. if the index goes up, you make money. If it goes down, you lose less than you would have without the floor.