There have been a few posts on this forum about people booking flights on budget airlines as if they are getting connecting flights, allowing only a few hours to make the 'connection'. The post usually continues with a comment about how critical it is to make the 'connection'.
The problem with this is that they aren't connections. Passengers have to retrieve and recheck their luggage. The budget airlines frequently have a separate terminal, which means another trip through security, too. If you are arriving on a large plane, it can take a fair amount of time just to debark.
And most important, since it is not a connection, there is no protection if you miss a flight because the first flight is delayed. That means that if your second flight is on the budget airline, your ticket is wasted and you will have to pay for a new one. If your second flight is on a major carrier, they will probably not automatically book you on the next available flight without charge, but will treat you by whatever policy is in place for missing a flight because you, for instance, overslept. The fact that another airline caused the problem is irrelevant.
I don't think using budget airlines to 'connect' to another airline, or sometimes even the same airline, is appropriate unless you allow for significant time between flights. The best thing would be to fly the first leg and at least stay overnight before getting the second flight. I certainly wouldn't do it for an international flight, where in addition to checking luggage immigration and customs must be dealt with.
I actually don't think using separately ticketed flights to make 'connections' is wise, regardless of the airline.