Mounted longbow

Insofar as I am aware the only difference between Mounted infantry and their non-mounted counterpart is their movement rate.

They fight the same and do not have to dismount for combat.

So yes to the distance shooting as well.

All those points mentioned by Joe are spread around the rules. Pat1958, you are not blind, sometimes the rules are obtuse.

This part is in Troop Definitions
"A very few armies have mounted infantry (prefixed by “Mtd”). These are on larger bases with their mounts, but remain foot and differ in that in good going they have the same tactical move as knights and can move more than once per bound."

The cryptic part "they remain foot" is supposed to indicate they act exactly like their foot type -- Mounted Bow act like Foot Bow except ...

This is in the Basing Your Models
"Mounted Infantry are based as 3-4 foot figures plus a vehicle, led mount or mounted figure. "

This is in the deployment section
"Cavalry, Light Horse, Camelry, Mounted infantry and Auxilia or Psiloi must deploy at least 2 BW away from battlefield side edges and others at least 4 BW away."

This is in Tactical Movement section
"3 BW If Knights, Elephants, Camelry or any mounted infantry and only in good going, or if “Fast” foot in any going."

This is in the second move section
"(a) Light Horse or mounted infantry; and making a second or third move that is entirely in good going."

This is in the Side Support section. You must infer that the rule applies to Mounted Bows if solid as they they remain foot
"“Solid” Bows add +1 if supported by “Solid” Blades."

In outcomes, another place for inference, for Mounted Lb and Mounted Cb
"If its total is equal to that of its opponent:
Knights or Camelry Destroyed in close combat by any Blades or any Bows that are Lb or Cb, " There are Mounted Lb and Cb

for example: IV/43b Hungarian Army for the Crusade of Nicopolis 1396 AD: 1 x General (3Kn), 1 x Hungarian nobles (3Kn), 4 x horse archers (LH), 2 x crusader knights (3Kn), 1 x crusader crossbows (Mtd-4Cb/4Cb), 1 x spearmen (Sp) or crusader ribauds (5Hd), 2 x archers (Ps or 3Bw).

This is in the Beaten section
"If its total is less than that of its opponent but more than half:
Destroyed if sacking a city or if defenders of a city, fort or camp or denizens or camp followers that have sallied or mounted infantry in bad going."

It is important to note that there are many references to "mounted," such as in combat factors and outcomes, but this does not refer to "mounted" infantry, which as noted in the first example, "remain foot."

Also note that while Phil uses the generic term "mounted infantry" there is only one type of infantry that is mounted, and that is Bows. There are types that dismount from being mounted to being foot such as 3Kn//4Bd. There are 3Cv//3Bw, but these are not mounted bow, these are cavalry that convert into bows -- mounted to foot.