There is a sort of FFP in League One and Two. I don't know how it pans out in reality but it looks more restrictive than the Championship on paper.
They call it SCMP, the salary cost management protocol. For League Two it's no more than 50% of a club's turnover to be spent on player related expenditure (wages and transfer fees), in League One it's 60%. There are certain things that count towards turnover, certain things that are exempt.
All that goes out of the window in the Championship under P&S (profit and sustainability). There's loss thresholds where clubs are allowed to lose a fixed amount rather than it being based on a percentage turnover. It's really complicated, reported over a rolling period and teams relegated from the Premier League get a little more leeway as it's assumed they'd be carrying a higher wage bill that they couldn't otherwise cut in time. It's probably why we end up with a lot of yo yo teams like Burnley, Sheff Utd etc. especially when you add in their parachute payments.
I think a club newly promoted to the Championship can incur greater losses in terms of number of pounds they could lose in League One typically speaking. The problem is they'd be going against teams with larger stadiums when it comes to matchday income, and larger commericial deals than they're currently facing in League One who are all allowed to lose the same fixed amount.
Still, it seems possible to take a smallish club all the way to the PL. Weren't Bournemouth backed by Russian investment? Not sure of the exact rules at the time though and if they had to comply to certain regulations or not. Still, they only have a 12k stadium and it didn't stop them. They're not historically popular so wouldn't have had the commercial deals of Sunderland, Leeds, Sheffield Wednesday etc. Wrexham probably have more commercial income than Bournemouth did at the time. Luton did it too with another similar sized ground to Wrexham, and I don't think they even broke the bank too much, not sure though.