When should "such forth" be used?

The idiomatic phrase is "and so forth" which is often also used as a part of the phrase "and so on and so forth". It is, IMO, an elided form of "(and so it goes on) and so it goes forth".

While "and such forth" appears to be reasonably commonly used (going by Google's results), this ngram suggests that it is at best a colloquialism or a dialectal peculiarity and at worst, a common error.

To answer the question in the title, I'd say that and such forth should never be used. The phrase and so forth should be preferred instead.

It would be understandable if you said “I can do some relocation and such” (with such acting as a pronoun, standing for “things like the one or ones already mentioned”), or if you said “I can do some relocation and so forth” (1, 2), but bundling the two forms together gives an untoward result.

So and such are merely variants of the same morpheme (like a and an, or he and him)

Such appears modifying (and quantifying) Noun Phrases, while so appears elsewhere. There are a number of idioms and fixed phrases that use them.

• It was so wonderful/such a good party that I died and went to heaven.
• So much/Such a large quantity of wheat was produced that they had to export some.
• He doesn't feel so depressed/such despair about it anymore.

In the context here, so forth is the idiom because forth is an adverb. Alternate idioms with such include such things and such stuff; I would normally consider such forth to be a nonce blend of those idioms.