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We have a great article that outlines the compatibility relationship between different versions of Tableau Server and Tableau Desktop. Have you seen it? Desktop and Server Compatibility | Tableau Software You are correct that older versions of Tableau Desktop can publish to newer versions of Tableau Server. For Tableau Desktop opening files, maintenance release versions (9.3.1, 9.3.2, 9.3.x) are always compatible with each other but minor releases (9.2, 9.3, 9.x) are not. One call-out that some people do not consider with different environments is that if a 9.2 Tableau Desktop user publishes their dashboard to a 9.3 Tableau Server, the workbook will automatically be upgraded to Tableau 9.3 and the customer will not be able to edit it in Tableau Desktop 9.2. Good news though: if a user needs multiple versions of Tableau Desktop, there is no issue with having multiple versions installed on the same machine.
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Edit: Strike out incorrect information.
As per Desktop and Server Compatibility | Tableau Software published workbooks arent automatically upgraded so can be downloaded and opened in their original Desktop version
"A workbook or data source with an extract created in Tableau Desktop 8.0 or later retains its original version when published to Tableau Server. For example, when you publish a workbook using Tableau Desktop 8.1 to Tableau Server 8.2 and later, the workbook remains a version 8.1 workbook and can be downloaded and modified in Tableau Desktop 8.1.
Prior to Tableau Server 8.2, publishing a workbook or data source with an extract to server automatically upgraded the version of the workbook or data source to the same version as Tableau Server."