Firstly: Continuity in Making up Fasts
Continuity in making up Ramadan fasts is not wajib. This is by agreement of the four schools of jurisprudence, and the position of the majority of scholars.
Secondly: Delay in Making up Fasts
1. The ruling of delaying making up fasts until just before the next Ramadan
It is permissible to have delay  Hastening to make up the fasts is better. in making up fasts, such that they are made up during any time of the year, as long as they are made up completely before the entering of the next Ramadan. This is by agreement of the four schools of jurisprudence.
2. Delaying the making up of fasts until the entering of the next Ramadan without an excuse
Scholars have differed over the one who delays the making up of fasts until the next Ramadan enters without a valid excise, and this is according to two views:
The first: is that he is obligated to make up the fast as well as pay compensation, which is the feeding of a destitute for each missed day. This is the position of the majority: Malikis, Shafi`is, and Hanbalis.
The second: is that he is obligated to solely make up the days. This is the position of the Hanafis, Ibn Hazm, al-Shawkani, Ibn `Uthaymin, and some of the Salaf.
3. The ruling of fasting voluntarily before making up wajib fasts
It is not wajib  Some of them have deemed it permissible, and others as makruh. to make up wajib fasts before fasting voluntarily, as long as there is sufficient time to fast both. This is the position of the majority: Hanafis, Malikis, Shafi`is, and a narration from Ahmad.