Alt + Drag a subgraph is corrupting the original subgraph

### Prerequisites

- [x] I am running the latest version of ComfyUI
- [x] I have searched existing issues to make sure this isn't a duplicate
- [x] I have tested with all custom nodes disabled ([see how](https://docs.comfy.org/troubleshooting/custom-node-issues#step-1%3A-test-with-all-custom-nodes-disabled))

### What happened?

Alt + Drag a subgraph is corrupting the original subgraph (or maybe both?)

### Steps to Reproduce

1. Select a subgraph.
2. Alt + drag the subgraph to create an independent copy.
3. Edit the new subgraph (e.g., delete some nodes that have input links).
4. Check the original subgraph, the nodes are still present (as expected), but their links are gone (not expected).

### How is this affecting you?

Feature doesn't work as expected

### ComfyUI Frontend Version

1.28.0

### Browser

Firefox

### Console Errors

```javascript

```

### Logs

```shell

```

### Additional Context

_No response_

┆Issue is synchronized with this [Notion page](https://www.notion.so/Issue-5716-Alt-Drag-a-subgraph-is-corrupting-the-original-subgraph-2766d73d365081ee86bcc4d1f0f01ca8) by [Unito](https://www.unito.io)