Since the transport sector accounts for one of the highest shares of greenhouse gases (GHG) emissions, several existing proposals state the idea to control the by the transportation sector caused GHG emissions through an Emission Trading Systems (ETS). However, most existing approaches integrate GHG emissions through the fuel consumption and car registration, limiting the tracing of emissions in more complex modes e.g. shared vehicles, shared rides and even public transportation. This paper presents a new design of a user-centric ETS and its implementation as a carbon Blockchain framework for Smart Mobility Data-market (cBSMD). The cBSMD allows for the seamless transactions of token-equivalent GHG emissions when realizing a trip, or an emission trading action as well as the transaction of individual, service or system-wide emission performance data. We demonstrate an instance of the cBSMD implementation for the transactions of an ETS where all travellers receive a certain amount of emission credits in the form of tokens, linked to the GHG price and a total emission cap. Travellers use their tokens each time they emit GHG when travelling in a multi-modal network, purchase tokens for a given trip when they have an insufficient token amount or sell when having a surplus of tokens due to a lower amount of emitted GHG. This instance of cBSMD is then applied to a case-study of 24hours of mobility of 3,186 travellers from Oakville, Ontario, Canada, where we showcase different cBSMD transactions and analyze token usage and emission performance.