How can you clear up a bench warrant in Alberta if you live in BC?

If the bench warrant is for an unpaid traffic or bylaw fine, or any other non-criminal fine, you can simply pay the fine to the Receiver General of British Columbia, which will cancel the warrant. Be sure to provide your full name and the ticket number.

If the bench warrant is for Failure to Appear, overdue Child Support, or other minor criminal or provincial offenses, you should engage an attorney who may be able to arrange a court date with the Crown and appear on your behalf without the need for you to travel to British Columbia. If you wish to plead guilty, you may request to have the charge transferred to Ontario. If you wish to plead not guilty, you will have to appear in B.C. to contest the charge.

For more serious summary offenses, and for any indictable offense, your attorney can arrange for you to surrender to the appropriate police force in B.C. Doing so in advance reduces the likelihood of you being denied bail, or spending several days in a Remand center waiting for a preliminary hearing.

A bench warrant is a warrant that is issued by a Judge or Magistrate pursuant to other proceedings. It is different from a warrant requested by the Crown or Police as part of a criminal investigation. Warrants can be "endorsed" or "unendorsed". This refers to whether the issuing Judge has signed the part of the warrant that allows you to be released by a peace officer on a recognizance, instead of having to stay in custody until you appear in court for a bail hearing.

Most bench warrants for Failure to Appear are unendorsed, meaning you will remain in custody until a Magistrate or Judge sees you and determines if you should be granted bail. This can be a few hours, or a few days. Most warrants for unpaid fines or other minor criminal offenses are endorsed, meaning you will be fingerprinted, formally charged, and then immediately released with a summons.

Note: While prosecutors or police forces may decline to act on extremely old warrants and charges, warrants for indictable offenses under the criminal code do not expire. There is no statute of limitations for indictable offenses in Canada. The statute of limitations for summary offenses is 6 months, however, that is only how long the Crown has to charge a person. Once charged, the file remains active until it is dealt with. You can be charged without ever having been arrested. In the context of this question, the subject of the warrant has generally already been charged, so the statute does not apply.