Leveraging an existing audience from other platforms can significantly aid in growing a Twitter fan base. The benefits are straightforward:

1. Cross-utilization: You can tap into a ready-made audience that already has interest in your brand or persona.
2. Efficient Time Usage: It saves time compared to starting from scratch on a new platform.

However, there are potential drawbacks:

1. Audience Divergence: The audience’s platform preferences may differ. Some might prefer the original platform and may not wish to transition to Twitter.
2. Content Dissonance: Different platforms often require distinct content styles, and cross-posting may not always resonate well with all audience segments.

To implement cross-promotion strategies without alienating audiences, consider:

1. Platform-Specific Content: Customize your content to suit the platform and audience behaviour on it.
2. Audience Segmentation: Identify which part of your audience is most likely willing to engage across platforms and target them accordingly.

To sustain audience growth and engagement through cross-channel promotion:

1. Consistent Branding: Ensure your brand’s voice and visuals remain consistent across platforms to help audiences recognize and engage with your content.
2. Regular Monitoring: Regularly analyse your audience behaviour and preferences using robust analytics tools on platforms like Flinque to effectively tailor your content strategies.

Finally, remember that the specific needs of your team or brand guide which platform is the best fit. Different platforms have different strengths and features designed to cater to varying use cases.