Praying salat al-qasr (shortening prayers) while on a journey is a sunnah (see for example in sahih al-Bukahri and sahih Muslim), some scholars tend to say one shouldn't forsake it. The hanafis even say it is wajib, so they won't allow it for a traveler to pray prayers fully (note they even don't allow them to join prayers, but both of this rules have no backup in the sahih sunnah). But one can pray the prayers fully - especially if the Imam is a resident - even if many scholars (especially from the hanbali and salafi school) dislike it, it is a known and accepted act performed by the sahaba (may Allah be pleased with them):
See for example in Muwatta' al-Imam Malik:
Ibn 'Omar used to pray qasr prayer on journey unless he prayed behind an Imam (see also).
'Omar and ibn 'Omar if on journey leading a congregation they would ask the (resident) people to continue the prayer.
If somebody who is on a journey would like to lead a prayer, he prays the shortened prayers and asks the residents to continue as quoted above.
But if the resident was the Imam it was -according this fatwa in Arabic - narrated from all the sahaba that they used to pray the prayer fully and this is - according to this fatwa (in Arabic) - the prevalent opinion. While the fatawa on islamqa say one can shorten the prayer see for example about shortening the prayers, and shortening behind a resident.
In the Maliki school it is even preferred not to lead a prayer except maghrib and sobh (fajr fard) neither for a non-resident for a resident, nor for a resident for non-resident. So unless the Imam is householder or an old or much honored person then if the Imam is resident, the non-resident should pray his prayer fully, if he is on journey then the resident should complete his prayer (see this fatwa in Arabic).
