Recently, self-supervised learning (SSL) from unlabelled speech data has gained increased attention in the automatic speech recognition (ASR) community. Typical SSL methods include autoregressive predictive coding (APC), Wav2vec2.0, and hidden unit BERT (HuBERT). However, SSL models are biased to the pretraining data. When SSL models are finetuned with data from another domain, domain shifting occurs and might cause limited knowledge transfer for downstream tasks. In this paper, we propose a novel framework, domain responsible adaptation and finetuning (DRAFT), to reduce domain shifting in pretrained speech models, and evaluate it for a causal and non-causal transformer. For the causal transformer, an extension of APC (E-APC) is proposed to learn richer information from unlabelled data by using multiple temporally-shifted sequences to perform prediction. For the non-causal transformer, various solutions for using the bidirectional APC (Bi-APC) are investigated. In addition, the DRAFT framework is examined for Wav2vec2.0 and HuBERT methods, which use non-causal transformers as the backbone. The experiments are conducted on child ASR (using the OGI and MyST databases) using SSL models trained with unlabelled adult speech data from Librispeech. The relative WER improvements of up to 19.7% on the two child tasks are observed when compared to the pretrained models without adaptation. With the proposed methods (E-APC and DRAFT), the relative WER improvements are even larger (30% and 19% on the OGI and MyST data, respectively) when compared to the models without using pretraining methods.