If you are using AX 2009 or AX 2012, your current infrastructure would most likely not be at par with the latest technology required to run the On-premise version of Dynamics 365. So, the right decision is to move to the cloud.
Further, moving to cloud by upgrading to Dynamics 365 Finance and Supply Chain gives lot of additional benefits such as new cross-platform UI (PC & Mobile, any browser), real-time BI, ease of access, reduced IT maintenance and upgrades.

Dynamics 365 can interface data with external applications using very common authorization mechanisms such as OAuth2 and data formats like JSON, XML etc.

Dynamics 365 can interface data exchange with external applications using very common authorization mechanisms such as OAuth2 and data formats like JSON, XML etc.

Dynamics 365 contains a user-friendly layer of data structure called Data Entities which facilitates easy building of reports and visualizations using Power BI. Dynamics 365 also comes ready with the tools ready to integrate with Azure ML / Azure AI services.

The approach to reuse the existing custom developed functionalities, reports and integration components depends on the complexity of the customization’s and the impact on the core processes. There are many functionalities and enhancements built into Dynamics 365 based on the industry needs. So, custom developed components of the previous versions need not be brought forward. However, if there are any specific developments you would like to be in Dynamics 365, those have to be developed. Most of the integrations may have to be reworked as Dynamics 365 uses OData connections, RESTful web services, JSON, DIXF.
There are many ISV solutions which might also be looked at for fulfilling your integration/customization needs.

Microsoft has done lot of functional enhancements and capabilities in many processes based on their R&D Dynamics 365 provides Real-time Business Intelligence, an in-memory BI solution that provides real time picture of how your business is performing, with new tools like Power BI for visualizing this data by creating dashboards for various business functions.
Lot of Platform innovations are also introduced by Microsoft so that you can take advantage of many new capabilities of Microsoft cloud, which includes common data model, now called Dataverse with Dynamics 365 and Office 365, you can use PowerApps for easily creating mobile apps, and Microsoft Flow for easily creating workflows, notifications etc.

Dynamics 365 has an on-premise version like previous versions of AX 2009 or 2012. With an on-premises deployment, existing data center investments can be leveraged and also, they can meet the regulatory and compliance needs, comply with data sovereignty rules in regions where there are no Azure Data Centers. However, some connectivity is required for system management and updates which are enabled through Microsoft Dynamics Lifecycle Services (LCS), a cloud-based application lifecycle management service.
Following are the requirements when On-Premise deployment is opted for.
- Configure your own high-availability and disaster recovery solutions.
- Stand up sandbox environments.
- Manage your own infrastructure, including scheduling operating system updates.
The additional costs to deploy and manage these capabilities might lead to higher infrastructure costs resulting in a higher Total Cost of Ownership (TCO). Tools for deploying Dynamics 365 and updates will be available to partners and customers via Lifecycle Services. However, unlike the cloud version, Advanced Analytics and Azure Machine Learning capabilities are not included in the on-premises deployment option.

The cost and timelines depend on your current AX system customizations, 3rd party integrations and modules implemented etc. To make a decision on upgrade, it is better to do a detailed analysis which includes a thorough study of your current processes and requirements on reporting and integration with other applications.