So what have been the major steps to enhance climate capacity-building?
Over the recent years, various arrangements have been established to enhance capacity-building under the UN climate change regime:

Credit: UNFCCC

1992, 1997: Capacity-building, the Convention & the Kyoto Protocol

Capacity-building is involved in many issues referred to in Convention articles, such as adapting to climate change, providing financial and technical resources, transferring technology, ensuring research, education and training, and improving national communications.
Specifically on education, training and public awareness, Article 6 of the Convention calls on governments to educate, empower and engage all stakeholders and major groups on policies relating to climate change – a call that is echoed in the Kyoto Protocol (Article 10(e)).

2001: Adopting Frameworks for Capacity-building

In 2001, the COP adopted two frameworks that address the needs, conditions and priorities of developing countries and countries with economies in transition (EITs). The frameworks are enshrined in decisions 2/CP.7 for Capacity building in developing countries (non-Annex I Parties) and 3/CP.7 for Capacity building in countries with economies in transition and are regularly reviewed. At regular intervals, the Subsidiary Body for Implementation (SBI) conducts comprehensive reviews of the implementation of the capacity-building frameworks in developing countries and in EITs. More information is available here.

2005: Parties to the Kyoto Protocol endorse the Capacity-building frameworks

In 2005, the Conference of the Parties serving as the meeting of the Parties to the Kyoto Protocol (CMP), at its first session, decided that the capacity-building frameworks are also applicable to the implementation of the Kyoto Protocol (decisions 29/CMP.1 and 30/CMP.1).

2011: The Durban Forum on capacity-building

At COP 17 in Durban, Parties established the Durban Forum on capacity-building. The forum meets once every year during UNFCCC negotiation sessions and is open to all stakeholders engaged in capacity-building efforts under the Convention. It contributes to monitoring and reviewing capacity-building efforts under the Convention, and focuses on the sharing of experiences, good practices and lessons learned in building the capacity of developing countries to mitigate and adapt to climate change.

2012: Doha work programme and the Dialogue on Article 6 of the Convention / Launch of the Capacity-building Portal

In 2012, the COP adopted the Doha work programme on Article 6 of the Convention and requested the SBI to organize an annual in-session Dialogue on Article 6 of the Convention for Parties, constituted bodies and other stakeholders to share their experiences and exchange ideas, good practices and lessons learned regarding the implementation of Article 6 of the Convention. A ministerial declaration was adopted at COP 20.
Also in 2012, the Capacity Building Portal was launched as an interactive tool that collects, compiles and disseminates country-driven information and allows the display of information from submissions of non-Party stakeholders providing capacity-building.

2015: Capacity-building in the Paris Agreement

Paris Agreement Article 11
Article 11 of the Paris Agreement defines capacity-building related goals, guiding principles and procedural obligations for all Parties. It calls on developed country Parties to enhance support to capacity-building in developing countries, and on developing country Parties to regularly communicate progress on implementing capacity-building plans, policies, actions or measures.
The Agreement also calls for country-driven capacity-building that is based on recipient countries’ needs and ownership. The Conference of the Parties serving as the meeting of the Parties to the Paris Agreement, at its first session (CMA 1) is to consider and adopt a decision on the initial institutional arrangements for capacity-building.
Paris Agreement Article 12
The Paris Agreement also recognizes the importance of climate change education, training, public awareness, public participation and public access to information, and asks Parties to cooperate in taking appropriate measures. 
Paris Committee on Capacity-Building (PCCB)
COP 21 established the Paris Committee on Capacity-building (PCCB) as a new constituted body, which aims to address capacity need and gaps, both current and emerging, and to further enhance capacity-building efforts. 

2015: Establishing the Capacity-building Initiative for Transparency (CBIT)

COP 21 also requested the establishment of the Capacity-building Initiative for Transparency (CBIT) in order to build institutional and technical capacity and to support developing country Parties in meeting the enhanced transparency requirements of the Paris Agreement. The GEF supports the operation of the CBIT.

2019: Review of the Paris Committee on Capacity-building

COP 25 concluded the first review of the PCCB. The COP decided to extend the PCCB for a five-year period until 2024 and to conduct a second review of the progress and need for extension of the PCCB at COP 30.

COP 25 provided clear guidance to the PCCB on the priority areas and corresponding activities for the next work plan period.

2019: Initial institutional arrangements for capacity-building under the Paris Agreement

CMA 2 agreed on the initial institutional arrangements for capacity-building under the Paris Agreement. The CMA decided for the PCCB to serve the Agreement, in line with its mandate and terms of reference established under the COP.

The CMA highlighted that capacity-building efforts and activities are undertaken by a range of constituted bodies under the Convention and the Paris Agreement, in line with their respective mandates, as well as by various bodies and actors outside the Convention and the Paris Agreement.