Reputation may not be everything, but it matters to Jesus. His concern was Kingdom reputation when the Apostles got involved directly in missional work, and when they were performing the Kingdom as an extension of his mission.
So, Matthew 10:11-13: here Jesus tells his missioners three things: (1) as a result of your Kingdom ministry, find a place to stay; (2) stay in that place until you leave that village; and (3) your peace-blessing will remain on a place or return to you.

Points #1 and #2 clearly pertain to reputation, just as the need to avoid the appearance of money-grubbing in the previous verses was concerned with reputation. The tendency must have been, as it was with some itinerants and mendicants, to start at one place and then if someone more wealthy offers his or her home, to move in there, and so on and on until one worked one’s way up the social ladder. Not so with Jesus. (This instruction is not unrelated to washing feet — it is about a servant attitude.) Jesus wanted his followers to do whatever they had to in order to avoid giving off the appearance of being poachers and money-seekers.
Many suggest the Lord’s Prayer is designed for such workers, with the bread request being a clear example of what needs to be said when you get up in the morning and are planning a day of visiting a new village with Kingdom news.
Also part of #1 is the need to discern who is “worthy.” Missional Kingdom activity is a personal thing: it finds people to trust.
Kingdom missional work offers the peace of Jesus to others — and this peace transcends inner peace as for Jesus “peace” pertains (see Matt 5:9; Luke 10:5) to the Kingdom condition of peace that is to permeate from Jesus through his followers into communities in order to transform life and society.
When we say “The peace of the Lord” perhaps we could think of Luke 10:5 and missional work and that we are graced by God to be empowered to offer the peace of Jesus to others — and it can stick!