It's 1580's England. I am an archer/longbowman, and I'm told that we are transitioning to firearms as our standard weaponry. What did other ranged soldiers of the time think about this, and what did the conversion process entail?

By that time period, most major continental European armies had phased out the bow and arrow. Rosters of the Army of Flanders from 1567 do not mention bowmen at all. It was already recognized that firearms were vastly superior.

English, Scot and Irish militaries should not be far behind in this realization, small numbers of them regularly campaigned on the continent as mercenaries or adventurers. So a bowman given an upgrade to firearms should be quite glad to be given the facility.

Going back a bit in time, even Henry VIII Tudor was already aware of the reforms undertaken by the armies of Charles V as they transitioned away from bows to firearms. By 1560 there were already about an equal number of firearms and bow among the shot footmen; by 1595 the bow was no longer considered acceptable for a first-line formation.

This is not to say that the bow completely disappeared, first we have to note that there was not yet one singular "state army" and therefore we have a variety of armies from the first-line troops to the local militia, who did keep various bows for longer. They were simply no longer a factor in non-trivial engagements.

Renaissance armies, 1480-1650. by George Gush.