Is it permissible for a congregant to pray with an intention different from that of the Imam, such as praying Dhuhr behind an Imam who is praying Asr? And is it permissible for the Imam, after the Tasleem, to follow his former congregant who is completing his prayer with the intention of a new prayer?

The preponderant view is that a difference in intention between the Imam and the congregant does not harm or prevent the validity of following (the Imam). This is the madhhab (definition) of Ash-Shafi'i and many verifiers. Thus, there is no harm for a congregant to pray Dhuhr behind an Imam who is praying Asr, nor is there any harm for an Imam to intend another prayer behind someone who is completing his prayer from those who joined late, because following someone who joined late is permissible according to the preponderant view.