I don't have a problem with reusing code so much as sharing code, that's when the problem starts. Each client should just get a fork of the code and then you start modifications. If a client wishes to upgrade their project you can look at merging in changes then. If they just need minor changes then you stick with the old version. That is, treat it like software with a distinct release for each client, and don't attempt to actually share one version of the code.
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I don't have a problem with reusing code so much as sharing code, that's when the problem starts. Each client should just get a fork of the code and then you start modifications. If a client wishes to upgrade their project you can look at merging in changes then. If they just need minor changes then you stick with the old version. That is, treat it like software with a distinct release for each client, and don't attempt to actually share one version of the code.