I realyzed that I can change the layer 2 algo so that RBF is no longer required.
Thanks for informative help anyway.

I hoped that RBF would work as follows:

Assume I have the first graph consisting of the three transactions in the mempool as described earlier.
In order to replace them, I would broadcast an "inv" message containing the inventories of the 3 RBF transactions (the second graph). Upon reception of this inv message, a node would know, that these three transactions come as a package. So when the node sends me the getdata message where he request the three messages. I send back those three messages with three distinct "tx" messages (There is no "txs" message where you can send bulk transactions). Now when the node receives the first RBF tx message, he would not yet reject it, even though it does not pay more fees than the sum of all transactions removed from the mempool, but first it would also evaluate the next two messages. Only then it would decide whether to reject the three RBF txs or not and it would not, because each RBF tx pays one sat/B more that the txs that are replaced. Does that make sense? Is that not how it should, or maybe even is, implemented?

Because Bitcoin is the most secure and most widely adopted PoW network.

Because that is how the consensus algo of my layer 2 protocol works. Every token votes for a layer 2 block. The more tokens there are, the more voting power you have, it's similar to PoW. So a layer 2 miner may want the vote with only one token, but he may also chose to increase his voting power by doing several tokens. If his tokens get not included in the next Bitcoin block however, he loses the fee he paid for the tokens, in this case RBF would be good.

No it's not. Because this is not about payments, these graphs are used for some layer 2 protocol. Each tx in the graph contains one OP_Return output plus a change addess. It's a token. The change output is used by the subsequent tx to make a new token. However, the tokens are only usefull for the layer 2 protocol, if they are inserted in the latest block. If the miner does not include them in the block, they sort of expire from the perspective of layer 2. Therefore they have to be RBF ed.

If I understand correctly, this is poses a problem for me. What I want to do is to replace all three transactions.
Let's say my first graph looks like this

TX₁₁ (9 sat/B) -> TX₂₁ (9 sat/B) -> TX₃₁ (9 sat/B)

Now I want this graph be replaced by this graph

TX₁₂ (10 sat/B) -> TX₂₂ (10 sat/B) -> TX₃₂ (10 sat/B)

For the miner it would make sense to replace the first graph with the second one since it pays 10 sat/B instead of 9. However, if I understand you correctly, when I broadcast the first RBF transaction TX₁₂, the miner does not yet know that I am up to also replace TX₂₁ and TX₃₁. So he wants me to pay a transaction fee of like 30 sat/B for TX₁₂ which is not what I want, since I intend to replace TX₂₁ and TX₃₁ with TX₂₂ and TX₃₂ as well.
Is there a solution to my problem?

I have three transactions that build a graph, i.e. the output of the first transaction is the input of the second transaction and the output of the second transaction is the input of the third transaction. RBF is enabled. All three transactions are still unconfirmed in the mempool. Now I want to replace all three transactions by a new graph of three transactions that each pay one satoshi per byte more fee than the old ones. Is it possible to replace all three transactions this way, or is it necessary that the first transaction of the new graph pays more fee than all three transactions of the first graph combined?

I ask this, because I have manually tried this with the Broadcast raw transaction online tool of the Blockstream website. I generate the raw transactions with my own program.
I was successfull to send the first three transactions of the first graph with a fee of 9 sat/byte. Then I wanted to replace the graph, so I incremented the sequence by one and increased the fee to 10 sat/byte and spent the same input that was spent by the first graph. But when I wanted to broadcast the first transaction I got the message, that the replacing tx does not pay enough fee, it should pay more fee, and the number it suggested was so high that it seems to me it wanted that the transaction paid a higher fee than the whole graph it opted to replace.

Now, I wonder if it would work, if I sent the new graph with a client directly to the network, where I could announce all three transactions of the new graph at once by an "inv" message, instead of using the online broadcasting tool where I can only send one tx at once.

If it does not work to replace a graph as I described then I think that would be a bug.

Please enlighten me