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Question: Taraweeh I attend does 20 Rakats but no Tahajjud and finishes sooner than another mosque which prays Tahajjud. Can I pray 20 Rakats at my mosque then attend the other mosque and continue praying the rest of their Taraweeh and then Tahajjud?


The two types of prayers you mentioned (i.e., Taraweeh as well as Tahajjud, also often called Qiyam al-Lalyl) are voluntary prayers entailing great rewards and blessings. 
The Prophet, peace be upon him, was asked about the manner of performing these prayers and his answer was: "The (optional) night prayers are to be prayed two rak'ahs and two rak'ahs; if you fear that dawn is approaching, then you should finish it off by praying Witr." (Reported by Bukhari)
It is clear from the above one is allowed to pray any number of rak'ahs since there is no fixed number prescribed. 
Therefore, you are allowed to go from one mosque to another and pray as many rak'ah as you wish as long as you have the energy and stamina to do that without sacrificing your other priorities in life such as performing your spousal duties, work to earn a living, etc. We may do well to remember the words of the Prophet, peace be upon him: "Your body has rights, your eyes have rights, your guests have rights, your spouse has rights: so give each one his or her due." (Reported by Bukhari).


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