Is it obligatory to make up for missed days from previous Ramadans in order sequential, or is it permissible to begin by making up for days from the more recent Ramadan first, in order to be able to fast the six days of Shawwal?

The questioner should know that they have a fixed number of days that they owe and have not made up. If the current Ramadan finishes and they have broken their fast on some days during it, they must combine these with the previous days and make up the total number of days as an obligation before the next Ramadan. They are not required to specify the intention for the day being made up. They must also pay the fidyah (feeding a poor person for each day they delayed making up the fast without a valid excuse). As for fasting the six days of Shawwal before making up the obligatory fasts, this is permissible. And if they are occupied with making up the fasts in Shawwal, many scholars have held the view that making up the six days of Shawwal is legislated.