The sheriff's department from the county that issued the warrant is responsible for the transport. That means deputies from that county will travel to wherever the inmate is currently being held and physically transport them back to the jurisdiction where the warrant originated.
Two things about how this works that families always want to know. First, they will not announce when they are coming. The pickup happens without advance notice to the inmate or their family, and that is entirely intentional. A scheduled transport creates opportunities for complications that nobody in law enforcement wants to deal with. Second, they are not going to forget. A warrant is an active legal instrument and the issuing county has every incentive to close it out. The idea that a jurisdiction might just let it slide and never come is not how this works in practice.
The timeline is the one genuinely unpredictable variable. Depending on the distance between facilities, staffing, transport scheduling, and how busy the issuing county's sheriff department is, a pickup could happen within days or it could take weeks. Smaller counties with limited transport resources sometimes take longer simply because they cannot dedicate a vehicle and two deputies to a long-distance pickup on short notice. But the warrant does not expire and the pickup does eventually happen.
If your inmate is waiting on an inter-county or interstate transport, the current holding facility can sometimes give a general sense of where things stand, though they will not have specific scheduling information either.