Introduction
Since the AGEOD system is different to many other approaches, I thought it would be useful to do a short series of how to guides on key game mechanics.
I'll start with 'march to the sound of the guns' (mtsg) as that is a key concept and one that those new to the system can be caught out by.
The basics
One of the good things about the AGE model is you can play very well by not delving in too deep into the game mechanics. The games put you firmly in charge of the operational side of the war and removes the tactical element.
So here's the very basic rules for mtsg.
1) An 'army' in Wars on Napoleon has at least one stack. This has the commander in chief and is marked by a little star. Here's Ney's army at the start of the waterloo campaign. It has two stacks (ignore the Lille garrison), his own 'army' (which in this case is just him) and a corps. The corps will flash red on the map when you select the commander.

2) An 'army' can consist of only the commanders stack. This is common early game for the Austrians etc as they lack the corps structure. You'll also see this on quieter theatres. In this case there is no scope for mutual support.
3) Only units in the same army can support each other. This is important. A corps from another 'army' will not support a corps from a different 'army'. So if we look at the full French set up we find they have 3 'armies'. So a corps from Grouchy's force won't support Napoleon.

4) What I would do is remove Grouchy's army and make him into a corps of Napoleon's.
5) Once you have the command structure working, mtsg might happen;
6) It can't happen in the first hour of combat (so your lead corps may have to fight alone for this period);
7) Each battle round (1 hour) the chance is tested again (so units can contribute in one hour and not in the next – you'll see this on the battle gauge as numbers fighting rise and fall)
8) its more likely to happen if:
The supporting stack is 'active' (ie its commander has no yellow envelope);
The quicker the distance between the two provinces, the more likely it is to happen (so a major road in flat terrain is ideal);
if the army HQ is either making the reaction (so its often an idea to keep this back) or is adjacent to the unit making the reaction;
better led forces (both immediate commander and commander in chief) are more likely to react.
Since an early game Austrian or Prussian army has no corps, it must move as a single block (or risk being defeated in detail). This is bad for various reasons including command, supply, attrition and movement speed. Operationally, in the early game the French have a huge advantage.
The details
You can find all the rules for this (and other parts of the combat engine) in the file 'GameLogic' in .../Wars of Napoleon/NGC/Setttings
if you don't like how any part works ... you can mod it (use notepad or similar).
So what are the detailed rules:
resBaseChanceOff = 100 // Base chance if in Offensive posture
resBaseChanceDef = 90 // Base chance if in Defensive posture
resCostPerDay = 10 // -10% for each day of marching
resCohCostPerDay = -1 // -1 cohesion for each day of marching
resModAdjGHQ = 10 // +10% if adjacent to army HQ
resModIsGHQ = 25 // +25% if the army HQ itself
resModLeaderStrat = 5 // +5% for each pt of strat factor of the leader
resControlChunkMod = 5 // Every 5% of MC lacking gives -1% chance (both for start and end region)
First rule is important. A force in an offensive posture has no malus, on one of the defensive has a 10% malus. So it maybe an idea to place reserve forces into attack mode, even if you are actually defending.
Second rule you lose 10% per day it would take to march to the battle province. Things that matter are the traffic penalty, weather, leader traits (some are slow or fast movers) as well as terrain. But if it would take 20 days to march to the battle province – it is really not going to happen.
Note you pay a cohesion cost for joining the battle, so your reacting forces are less fresh than those who start the day on the battlefield.
Third rules shows the importance of the army HQ. Corps in an army will support even out of HQ range, but if they are adjacent to the HQ then their chance to react improves by 10%. The HQ itself has a 25% bonus, so this may affect how you organise your armies.
Fourth rule is a bonus for the strategic factor. Remember this is the base competence of the commander plus that of the CinC (if in the same theatre). So Napoleon leading say Davout will give you near certainty of a response, some early game Prussian numpty may well decide not to bother. As you can see in the Waterloo campaign, both Ney and Grouchy are less than wonderful (3 for strategic rating). Another reason for an early re-organisation in my opinion as bringing them under Napoleon's direct command means they benefit from his ratings.
Final rule relates to the amount of military control you have. In effect if in hostile terrain, messengers etc may go astray or be killed.
Add up the various positive and negative measures, if it reaches 100, then the force will march off to the battle. If its below 100% then that indicates the chance of a reaction.
If you want even more details, can only suggest reading the AGEWiki. An invaluable resource, especially if you want to mod AGEOD games.