We can be inclusive of those to whom the issues of the environment and homosexual rights are important.
Re: the environment, Republicans care as much about clean water, air, etc. as the libs, we just don't want business to be crippled by draconian and unnecessary laws. It's possible that some can be convinced of that, when the issue is explained.
Re: homosexual rights, Republicans think that everyone should have the same civil rights, and that there shouldn't be any special rights for any group. There are some homosexuals who agree, and who don't go in for activism. They simply want to be left alone. Why not encourage their participation in Republican issues?
On neither set of issues am I assuming any change of principle on the part of the Republicans, just the attempt to convince voters who haven't supported Republicans in the past.