After you have determined, together with the ISL lawyer and, where applicable, a lawyer in a third EU Member State, the best place for your client to be prosecuted, you should consider whether it is appropriate to make representations to the relevant authorities.
Since the CJ Framework Decision does not provide binding criteria on how the jurisdiction should be chosen, and Article 50 CFR has not been interpreted as proscribing the existence of parallel proceedings, ultimately the decision is one for the national prosecuting authorities, or the competent court dealing with the criminal case, in the relevant Member States. This will be determined in accordance with their national law and practice. If no agreement can be reached between them, they will all proceed with their cases and the decision that first becomes final will prevail, pursuant to Article 54 CISA and Article 50 CFR.
But since EU law does lay down information and consultation obligations for the Member States, as soon as parallel proceedings are detected, you should consider whether to invoke these provisions in an attempt to trigger consultation proceedings (Articles 5 to 13 CJ Framework Decision 2009/948/JHA together with Recital 5), or whether to let parallel proceedings continue and seek a final decision to be reached first in the most favourable Member State for your client.
It may be difficult to convince the relevant authorities to proceed in a Member State that is still investigating a case if another Member State has already issued a formal indictment, is already at the trial stage, or has progressed even further.
The financial capacity of your client to fund a cross-border dual or multiple defence team to defend him simultaneously in the multiple parallel proceedings may also have a bearing on whether the conflict issue should be brought to the attention of the authorities.
If the conflict arises during EAW proceedings and you conclude that the most suitable jurisdiction is the Executing State, then you may have no choice but to argue the issue in order to obtain a decision not to surrender. If the person is surrendered, it is highly likely that the Executing State will have waived jurisdiction (or made use of a similar mechanism) in favour of the Issuing State. The same applies if the conflict is between the Issuing State and a third EU Member State. If you have concluded that the latter would be the best place for your client to be prosecuted, together with the ISL and third Member State lawyer, they should trigger consultation proceedings in their Member States, which could ultimately lead to the EAW being withdrawn, should the Issuing State waive jurisdiction (or use a similar mechanism).