Not all effects need to scale. Any boost to a d20 roll (advantage) or status effect (prone) is just as good at level 1 as it is at level 20.
Others will probably scale by themselves. Giving allies THP equal to twice your warlord level, once per short rest. Possibly something with your proficiency bonus, or based on your Int or Cha.
But mostly I imagine simply getting more of them, like the warlock. So you can stack push and slow on your eldrich blast advantage and avoid OA's against the target if your attack.
Maybe even let you take the same one multiple times, so you can give THP twice per short rest.
Scaling on it still seems like a mess
I'll ask basically the same thing of you that I did of Tony. Let's take a level 5 warlord.
Give me an example of a balanced short rest, an example of a balanced long rest and an example of a balanced at will ability for level 1. I'm assuming you will grant a character 2-3 invocations at level 1?
Level 3 is generally a considerable power bump for PC's. Will you grant more of the level 1 invocations? Will the level 1 invocations scale well enough to still be strong enough at level 3? Will there be special level 3 requirement invocations that are stronger than level 1 invocations that you now can choose from? Do you not scale invocations at level 3 at all? Maybe some other mechanic grants the warlord his boost at level 3?
Most importantly, say I pick 2-3 at will invocations at level 1. They won't be able to add very much at level 1 or they will be too strong. It's going to be difficult to scale them at level 3 since there's probably not even a dice worth of anything we can look to double. It would take quite a few at will "balanced level 1 invocations" to be added in at level 3 to make level 3 be balanced with anything.
I'm worried you are trying to use a mechanic as the primary warlord mechanic that isn't going to be scalable like we need it. Warlocks get invocations and get spells that upgrade just like a caster. Could you imagine trying to make their invocation system work if they didn't get regular spell slots at all and everything was just an invocation? We would be asking the same questions. How many invocations do I get. When do we get upgraded invocations either through auto upgrading by level or through access to stronger ones by level.
Possibly tiered invocations at level 1,3,5,7,9 etc. At level 1 you choose 2-3 invocations. At level 3 you may choose 2 level 3 or lower invocation. At lever 5 choose 2 level 5 or lower invocations. Etc.
That's going to be a lot of invocations by max level but it might could work? I still don't like it as a lot of 1 use abilities that you can't trade for a different 1 use ability feels pretty bad and is likely the reason they went away from vanician casting. I don't think we need a vanician warlord... lol
Any other implementation ideas?