When will solar eclipses become lunar transits? When will the moon be far enough away from the Earth that we would no longer consider it a 'solar eclipse', but rather a 'lunar transit' when the moon travels between the Earth and Sun?
When looking at definitions of [transit] and [eclipse] or more general [occultations] it is noteworthy that the degree of covering is not defined, so technically speaking each eclipse is also a transit and an annular eclipse even more so.

Details for next eclipses can be found at [this NASA page]. The next annular eclipse, err, transit happens next Sunday 2017-02-26 and is visible in South America, south Atlantic and Afrika.

As you were referring to the [increasing distance between earth and moon] (+3.8 cm /yr), it then depends on an exact definition on when an eclipse is not regarded as an eclipse anymore. Assume for simplicity that this is the case, when the diameter of the moon has halved (one quarter the area), this means that the distance had to be doubled. 
This would take [380 Gm] / 3.8 cm/yr = 10 Tyr (using SI prefixes). 

Note that one might need to take other effects into account, too, for example the [increasing distance of earth and sun] (~15 cm / a) and that the [sun is becoming a red giant] in a couple of Gyr. 

Sorry, would have loved to post more links, but are not allowed to currently.