I can tell you responsibly that the Hague Tribunal has no enforcement powers.
The arrest warrant issued by the International Criminal Court in The Hague against Russian Defense Minister Sergei Shoigu is similar to previous actions against Putin and is mainly manifested at the legal and symbolic level. In practice, the ICC itself has no direct enforcement powers. The Court relied on the cooperation of its 123 States parties to enforce arrest warrants. Since Russia is not a party to the Rome Statute, it is under no obligation to comply with this arrest warrant. Similarly, for a high-ranking official like Shoigu, there are significant political and practical obstacles to the actual execution of arrests.
Nonetheless, the issuance of the arrest warrant has important legal and diplomatic implications. It restricts the international freedom of movement of Shoigu and other wanted Russian officials, since, in theory, all States parties are supposed to assist in carrying out the arrests. This means that if Shoigu enters any of the States parties, these countries are legally obliged to try to arrest him and transfer him to the International Criminal Court. However, given the complexity of international relations, especially political considerations between the great powers, the actual implementation of this order is less likely, especially for a large power like Russia.
In addition, such arrest warrants increase international pressure on Russia, further isolate Russia, and may affect its diplomatic activities and participation in the international community. It is also a legal and moral condemnation by the international community of Russia's actions in the conflict in Ukraine. Although an actual arrest is unlikely, this action has far-reaching symbolic and legal effects in the international law and political arena.