Late at night rambling predictions...

subject to change, terms and conditions apply, may cause drowsiness or irritation

Travel: Jump drive technology developed by Human companies who didnt want to put all their eggs in the mass effect drives basket - recognising that for off grid/aka relay network exploration they might need something different and some conspiricy theorists were also concerned about eezo running out, being under the control of 'aliens' or just not being available if they wandered too far off the known galactic pathways (jump drive supplemented with an eezo fuelled engine for local travel)

Why are they leaving version one: The drive for the unknown amongst some humans and particularly wanting to 'discover' new parts of the Galaxy and 'be the first.' The kind of human drive that led us to mars and the mass relays in the first place.  Even with the known galaxy to continue to explore the human drive for what's over the next horizon is strong

Why are they leaving version two: From the first knowledge gained about some form of indoctrinating alien race (aka the encounters with artifact on Shanxi adn the meta-turians) certain black op elements of the Alliance Nacy and Systems Alliance leadership decided that they would need a plan B for either defeating or surviving this threat.  For many years the threat was focused on the threat of 'suprer Turians and the Ark projects were a by product of this ongoing work and was another 'need to know' operation like the one developing AI's

Why are they leaving version three: Once the Reaper threat was known (aka sovereign), despite being disavowed publicly, Human's (and other species?) decided to come up with a safety net in case things went pete tong... we heard Tevos say as much but much later when Thessia was invaded

When do they leave version one: It should be after the Reapers arrive so it gives a real impetus to the journey and offers the only explanation (well that I can think about without... you know... artistic licence) as to why they would head for a different galaxy and not just head off into parts unknown, off relay, aka most of the rest of the milky way

When do they leave version two: At some point before the Reapers actually arive and the travellers are sold a pitch that this is just a huge exploration/settlement expedition (aka the manswell expedition) but this is humanity's lifeboat

Why Andromeda:  Coz its a great name... not the milky way... big... very big... and will eventually bring everyone home when it collides with the milky way sometime in the future

[of course our sponsers would like to point out that this is all speculation and that's fine as this is all we have until N7 day (counting the minutes) and if you want the lazy version see below in the small print... but thats not as much fun]

Disclaimer:
Travel will probably be provided by a form of hithertoo unknown methods of using FTL drives without the need to discharge or harnessing relay technology to the arks
Why they are leaving is because the milky way could no longer sustain the split personality and space time schism that it now resides in due to.... not gonna say it but it rhymes with bending...
When... oh well that will depend on how much of the Trilogy they want to harness as back story

Final message for the Mass Effect Milky Way from Bioware 'you existed because we allowed it and you ended because we demanded it'
End transmission