Yes. A grappled grappler cannot move.
When you want to grab a creature or wrestle with it, you can use the Attack action to make a special melee attack, a grapple. If you’re able to make multiple attacks with the Attack action, this attack replaces one of them.
The target of your grapple must be no more than one size larger than you, and it must be within your reach. Using at least one free hand, you try to seize the target by making a grapple check, a Strength (Athletics) check contested by the target’s Strength (Athletics) or Dexterity (Acrobatics) check (the target chooses the ability to use). If you succeed, you subject the target to the grappled condition (see appendix A). The condition specifies the things that end it, and you can release the target whenever you like (no action required).
Escaping a Grapple.
A grappled creature can use its action to escape. To do so, it must succeed on a Strength (Athletics) or Dexterity (Acrobatics) check contested by your Strength (Athletics) check.
Moving a Grappled Creature.
When you move, you can drag or carry the grappled creature with you, but your speed is halved, unless the creature is two or more sizes smaller than you.
- A grappled creature’s speed becomes 0, and it can't benefit from any bonus to its speed.
- The condition ends if the grappler is incapacitated (see the condition).
- The condition also ends if an effect removes the grappled creature from the reach of the grappler or grappling effect, such as when a creature is hurled away by the thunderwave spell.
After the first grapple, Andre's speed is still 40' but only 20' while dragging Conan. Conan is grappled so his speed is 0 (but nothing else happens). On Conan's turn he may use one of his attacks to grapple Andre back - Conan's Str(Ath) vs Andre's Str(Ath)/Dex(Acr). If successful, Andre's speed will now be 0 as well. This should prevent Andre from continuing his drag.
Andre can spend his action to try to break Conan's grapple, Andre's Str(Ath)/Dex(Acr) vs Conan's Str(Ath). If Andre succeeds, then he breaks Conan's grapple while maintaining his own. He can now continue dragging Conan. If he fails, he cannot move and his action is spent.
- A prone creature’s only movement option is to crawl, unless it stands up and thereby ends the condition.
- The creature has disadvantage on attack rolls.
- An attack roll against the creature has advantage if the attacker is within 5 feet of the creature. Otherwise, the attack roll has disadvantage.
If Conan is prone, he cannot stand up since his speed is 0. However, his grapple attack does not suffer disadvantage because grapple is a special attack - it actually results in a contested ability check not an attack roll.
Shoving a Creature
Using the Attack action, you can make a special melee attack to shove a creature, either to knock it prone or push it away from you. If you’re able to make multiple attacks with the Attack action, this attack replaces one of them.
The target of your shove must be no more than one size larger than you, and it must be within your reach. You make a Strength (Athletics) check contested by the target’s Strength (Athletics) or Dexterity (Acrobatics) check (the target chooses the ability to use). If you win the contest, you either knock the target prone or push it 5 feet away from you.
Shoving might still be a good option.
Conan could just shove Andre away (Andre's grab reach is only 5' AFAIK despite his club attack having reach 10'). Conan can use an attack to shove Andre 5' away, breaking the tackle. If successful, he can now move normally (standing up first if prone).
Conan can use his 2 attacks (we'll presume he is level 5) to both Shove(prone) and Grapple. However, even if the Shove(prone) is successful, Conan would not have advantage on his Grapple (or a Shove(away)).