You have no legal duty to transport your step children other than in the event of an emergency - where failing to transport them would endanger their health or safety. The obligation of transporting the children should be addressed by the parenting provisions of whatever Court Entry addresses the parenting time of the parents. If there are no provisions there, there may be a local rule that is applicable.
You may, or may not want to help facilitate the exchanges of the children. This is not a legal issue, but would be resolved based upon a number of other, personal issues between you, your spouse and the children. Usually, if there is an issue, it is that the other parent does not want the step parent having anything to do with the children. Since that is not the issue, if you become involved, it is because you choose to, not because you are legally obligated to do so.
IMPORTANT LEGAL NOTICE: Mr. Piper's response set forth above is not legal advice and it does not create an attorney-client relationship. Mr. Piper's responses to all questions posted on AVVO are intended to provide general information based upon the his understanding of the facts stated in the question, and are for the general educational purposes of the public, not any specific individual. Also, a particular case may involve additional facts and circumstances which might invalidate some or all of the concepts provided in this answer and therefore you should not rely upon this answer in any individual situation. In order to offer legal advice about this or any similar situation, a qualified attorney would likely need to consider many factors not stated in the question and would need to question the potential client in order to clarify the specific facts operable in that case. If you would like to obtain specific legal advice about this issue, it is recommended that you contact an attorney in your state. Mr. Piper is licensed to practice law in the State of Ohio, and may be contacted directly via email at: firstname.lastname@example.org.