How to create strongly typed langchain Runnables and pass data through several steps?

Question

The latest langchain LCEL enable us to create Runnables. Runnable abstraction can be used for a lot of things, even outside of chains or prompts.

In my scenario, as part of the chain pipeline, the first steps are not LLM or Prompts.

I want that every step (Runnable) receives a strongly typed input named ElementSelectionContext, but they can output different data. As part of the pipeline, between every step, I need to update the ElementSelectionContext fields with the output of the previous step, and pass it to the next step.

Enough talking, here's the code attempt

class ElementSelectionContext(BaseModel):
    element_name: str = Field(frozen=True)
    objective: str = Field(frozen=True)

    page: Optional[Analysis] = None

class ElementSelectionPipeline(RunnableSerializable[ElementSelectionContext, Optional[Element]]):
    def invoke(self, input: ElementSelectionContext, config: RunnableConfig | None = None) -> Optional[InterestingElement]:
        page = Page()
        exact_match = ExactMatch()
        pipeline = (
            page
            | {"page": RunnablePassthrough()} # WHAT do I add here to update the context and pass it through the next step??
            | exact_match
        )
        return pipeline.invoke(input)

class Page(RunnableSerializable[ElementSelectionContext,Analysis]):
    def invoke(self, input: ElementSelectionContext, config: RunnableConfig | None = None) -> Analysis:
        current_dir = os.path.dirname(os.path.abspath(__file__))
        file_path = os.path.join(current_dir, 'sample.json')

        with open(file_path, 'r', encoding="utf-8") as file:
            data = json.load(file)
            response = Analysis(**data)
            return response

class ExactMatch(RunnableSerializable[Analysis, Optional[Element]]):
    def invoke(self, input: ElementSelectionContext, config: RunnableConfig | None = None) -> Optional[Element]:
        #print(input)
        # ERROR in the next line since the 'input' is a dictionary, not 'ElementSelectionContext'
        first_value = next(iter(input.page.map.values()))
        return first_value

# USAGE EXAMPLE
pipeline = ElementSelectionPipeline()
response = pipeline.invoke(ElementSelectionContext(element_name="joba", objective="joba"))

1. In the ExactMatch the input is a dict and not ElementSelectionContext.
2. In the ElementSelectionPipeline I do not know how to update the ElementSelectionContext instance and pass it through the ExactMatch.

Any insights?