It's common for the language of an area to have replaced an older language previously spoken there. The new language typically will have a few words which can be traced to the old language, the latter being called a "substrate". For instance, Romanian has several words believed to be from Dacian.
But are there known cases of a language taking on traits pertaining to grammar – or anything besides lexical items – from the substrate?
Theo Vennemann claims that this is, in fact, what happened to English. According to him,
dozens of Celtic features have been identified in English, that which has been most discussed being the early rise of the progressive aspect, the difference between Peter works and Peter is working, which no other Germanic language has developed as early and as thouroughly as English and which is formally and functionally the same as in Welsh.
These would be specific cases of a more general phenomenon, summarized by two rules:
Superstrate rule: Superstrata give words to their substrata, less so structure.
Substrate rule: Substrata give structure to their superstrata, less so words.
The Balkan sprachbund is a group of languages which have interspersed grammatical features among each other. The concept of "superstrate" and "substrate" is less than clear here, since that implies a definitive movement from one language to another rather than the (sometimes uneasy) long-term coexistence of Balkan languages. But it is an example of fairly extensive "horizontal" (in the taxonomic sense) structural influence.
The terms "substrate" and "superstrate" are usually used in reference to Creole formation. Superstrate languages are the languages of the colonisers (so usually French/English/German) and the substrate languages are the local indigenous or imported labourers. In the traditional view of creolisation it's usually the superstrate language that provides the lexicon and the substrate languages that brings many grammatical structures. See the wikipedia article on Creole languages:
Most often, the vocabulary comes from the dominant group and the grammar from the subordinate group, where such stratification exists.
I know it's not exactly the same situation that you're referring to, but it gives an indication that substrate languages can influence language in more than just the lexicon.
This article from Britannica talks about Indo-Aryan languages borrowing grammatical features from Dravidian languages, and Dravidian languages borrowing vocabulary from Indo-Aryan languages. Sadly, it does not give examples of what it's talking about.
This paper supplies relevant examples, but I have not read it myself, and don't know how good or bad the paper is.
Background info: Indo-Aryan and Dravidian are the two major language families of India.
There are a few examples of substrate influence in the varieties of English in the UK. For a phonological influence, you can look at how old-fashioned speech in some communities of Wales, where Welsh is still widely spoken. The speech here, especially of older people, can maintain features like rhoticity (not found in most of Wales, but present in Welsh) and also being devoiced, so that older speakers may have cars pronounced with [s] rather than the standard [z], probably due to the absence of such a distinction in Welsh. As for grammatical functions, you could look at Shetland Scots, where Norn, an extinct Norse language has a substrate influence. For example, Shetland Scots maintains a T-V distinction, with two different forms for the formal and informal form of the second person personal pronoun.
Prepositions and subordinators (aka. subjunctions) are readily borrowed into other languages provided that they lack them, and such borrowing sometimes also pulls in the necessary syntactic structures. Several Southern American languages have borrowed subordinators from Spanish, and the idea of an explicit subordinating marker with them. Hm, I read this recently, from Cristofaro (2003) "Subordination" possibly?
