Purpose: Plena Inclusión is the National Federation of Organisations of People with Disabilities in Spain providing services, defending rights, and being an agent of social change. Families who have children with early childhood intervention (ECI) support needs have experienced a changing situation during confinement due to the pandemic. Some of these changes have critically affected some families, whereas others have been able to rely on professional support that has allowed them to develop strengths and improve their functioning as a family. This objective is, precisely, one of the keys to the Family-Centered Practice (FCP) model in ECI. This study aims to explore the narratives of families who have received support in ECI practices during the lockdown in the region of Castilla–La Mancha (Spain) within the frame of Plena Inclusión services. These practices have been directed, in many cases, toward tele-intervention. Method: Data have been collected through ethnographic interviews with families and have been processed through thematic analysis. Results: The results show the perception of families articulated in two thematic network focuses: the personal and family emotions regarding the past and present moment (fear, loneliness, despair, nervousness, uncertainty, anguish, and frustration) and the emotions experienced also in the present but projected toward future (feeling of competence, hope, adaptation, tranquility as they felt supported by professionals, relief for not being alone, trust in their process, or empowerment, among others). Conclusions: In parallel, focus groups with professionals who support these families were carried out, although the results of this analysis will be presented in future reports. The study opens new lines, which evidence the difficulty of the process of transformation toward actual FCPs.