Also keep in mind that there is a "First Touch" attribution model as part of Attribution IQ (Report time attribution). This works a bit differently from the evar concepts so far discussed.
From this lens, the date range of the report can come into play, depending on your lookback window (visit/visitor). There is a good example of first touch attribution for attribution IQ here: Attribution IQ
First-touch:- The first marketing channel to engage a visitor.Technically, the first-touch channel is an eVar with original expiration.
This will not change the source in first-touch until this will not expire the Adobe Analytics cookie or time range of first-touch is exist in the cookie. So first-touch will remain Search engine it doesn't matter after that you came through banner or etc..