Unless you Ready an action or are using optional rules, you cannot stop the creature from falling.
If a creature starts falling while another creature watches, the first thing we need to know to determine if they can do anything is how fast the creature falls.
You fall all the way down immediately
The default rule in the PHB (p. 183) doesn't specify exactly how fast a creature falls.
However, using the clarification from the preface to the optional rules on falling in Xanathar's Guide to Everything makes this default rule much clearer.
The [PHB] rule for falling assumes that a creature immediately drops the entire distance when it falls. (XGE, p. 77)
So, using the default rules as clarified by XGE, no creature has an opportunity to do anything once a creature starts falling. They immediately fall the entire distance and take however much damage or other effects they have triggered.
This means that if your Aasimar sees another creature falling, they have no time in which to do anything since the creature plummets immediately to the ground.
There are two ways you could act before the creature hits the ground
There are two exceptions to this in this situation:
You are using an optional rule and falling from over 500 feet
XGTE provides an optional rule for falling from a great height that adds a bit of realism:
When you fall from a great height, you instantly descend up to 500 feet. If you’re still falling on your next turn, you descend up to 500 feet at the end of that turn. This process continues until the fall ends, either because you hit the ground or the fall is otherwise halted.
In this case, if your falling creature falls from more than 500 feet above the ground and you are using this optional rule, then the Aasimar could, during its turn, try to help the falling creature.
If the Aasimar had readied an action to do something to interfere with the fall contingent on the creature starting to fall, then the creature falling would trigger that action. However, you can only Ready movement or an action, not both. So unless the creature is falling right next to them and/or your DM rules that helping to stop them from falling is not an action, this really won't help in this specific scenario.
As @PinkSweetener pointed out, the Aasimar could try using their Readied action to just fly underneath the creature and try slowing them physically with their body.
What happens if they make it to the falling creature?
There are actually no rules for having another creature use their flight to prevent another creature from falling. So this will be a DM call to decide how that works.
One way it could work is a grapple check to grapple the falling creature and then simply hold onto them. No further ruling about continuing to fall or splitting damage need happen. However, a DM may of course implement such things if they want. But there are no rules that talk about this scenario so it will be completely up to them.
What if this happens outside of combat?
The above assumes that all of this happens within combat or that the DM decides to run it in the structure of combat. This is generally a good assumption since pretty much all of the rules in D&D assume combat rules are in place.
However, if this happens in a case where this happens outside of combat and the DM chooses to run it outside of combat rules, they can do so. However, there are absolutely no rules for this and it basically goes right back to the case where the Aasimar reaches the falling creature in combat.