The Rules as Written Prevent It…
A lot of grapple fun is quashed by a creature being unable to take free actions (except, most of the time, speaking) when it's not the creature's turn. It seems like—but I'll add it anyway, just in case—that you already know that, like it says here, "Grappled creatures cannot make attacks of opportunity."
Finally, "If you successfully grapple a creature that is not adjacent to you," it says here, "move that creature to an adjacent open space (if no space is available, your grapple fails)." This is a radical change from D&D 3.5 where the grappler enters the grappled creature's square.
…But House Rules Sometimes Allow Free Actions Off-turn
Then all bets are off, and it's a whole new game for your eidolon-wearing buddy.
Complete Warrior includes a mancatcher, an exotic weapon that can automatically grapple targets and has special rules for grappling at a distance (since it has reach).
It has always been my impression that, the mancatcher aside as a special exception, grappling required an unarmed strike, and could not be performed without a free hand. The definition of handedness notes that
A light weapon [...] can be used while grappling,
which strongly suggests that other weapons, like a lucerne hammer, cannot be. This is referring to the rule under grappling for attacking an opponent, which says
You can make an attack with an unarmed strike, natural weapon, or light weapon against another character you are grappling.
It also says
You can’t attack with two weapons while grappling, even if both are light weapons,
which again suggests what I was saying earlier about unarmed strikes, as does the simple fact that successfully grappling deals damage as if you had performed an unarmed strike.
Nonetheless, I cannot find any explicit mentions of a free hand being required for grappling, despite being fairly confident they existed. I will continue searching for that, but in the meantime, I only have this circumstantial evidence. After all, the mancatcher does more than just allow you to grapple with it, since it has automatic grappling and a fancy trip-like grapple option, and the “can be used in grappling” might just refer to attacking with the weapon, rather than just starting the grapple. So all this evidence is circumstantial, and could be explained a different way.
But if you are convinced, please note (as @HeyICanChan reminds me) that a character can perform an unarmed strike without necessarily using their arms or hands, so they can do it despite having weapons in hand. As @WannabeWarlock points out, this isn’t at all unusual in real life, either—leg-based takedowns while pummeling with arms and fists is a pretty typical move in mixed martial arts, for example. Doing it with armor and a big weapon in hand, okay, that’s harder, but these are fantasy heroes, so whatever. But the big, important thing here is, even if you do this, you are still using an unarmed strike. That means none of your weapon’s properties—like reach—apply.
If you are unconvinced, then the rules basically don’t really cover grappling-at-a-distance, so you’ll still have to houserule something. I suppose you could use the mancatcher rules, if you want, but that would seriously dilute the mancatcher’s value in my mind. Or you could just use the rules as-is, and ignore the weirdness where, for example, your character just instantly moves into the target’s space regardless of the distance between them.
Best Answer
Melee isn't limited to or defined by a list of mechanical actions. Creative maneuvers in combat are not only possible, they're expected by the game and it provides rules for them. Flip to page 193 and read the sidebar — it's a core rule, notice, not marked as a variant or optional rule.
To reposition an enemy by flipping them, you just need your DM's buy-in that it's possible or reasonable (because ruling on that sort of thing is the DM's job), and then they and you will use the Improvising an Action rule to resolve it. It's a simple process: